<PAGE>   1





                           EXCHANGE AGENCY AGREEMENT


   
         This Exchange Agency Agreement (this "Agreement") is entered into as
of August 4, 1998, between IBJ Schroder Bank & Trust Company, a New York
banking corporation, as Exchange Agent (the "Agent"), and Thermadyne Holdings
Corporation, a Delaware corporation (the "Company").
    

   
         The Company proposes to exchange $1,000 principal amount of the
Company's 12 1/2% Senior Discount Debentures due 2008 (the "Exchange
Debentures") in exchange (the "Exchange Offer") for an equal aggregate
principal amount of the Company's outstanding 12 1/2% Senior Discount
Debentures due 2008 (the "Old Debentures") to the holders thereof (the
"Holders"), pursuant to the Prospectus dated as of August 4, 1998 (the
"Prospectus") and the accompanying Letter of Transmittal.  The Exchange Offer
will terminate at 5:00 p.m. New York City time on September 3, 1998, unless
extended by the Company in its sole discretion (the "Expiration Date").  The
New Debentures are to be issued by the Company pursuant to the terms of an
Indenture dated as of May 22, 1998 (the "Indenture"), between the Company and
IBJ Schroder Bank & Trust Company, as trustee (the "Trustee").
    

         Subject to the provisions hereof, the Company hereby appoints and the
Agent hereby accepts the appointment as Agent for the purposes of receiving,
accepting for delivery and otherwise acting upon tenders of the Old Debentures
in accordance with the form of Letter of Transmittal attached hereto (the "LT")
and with the terms and conditions set forth herein and under the caption "The
Exchange Offer" in the Prospectus.

         1.      The Agent has received the following documents in connection
with its appointment:

                 (a)      the LT;
                 (b)      a form of Notice of Guaranteed Delivery; and
                 (c)      the Prospectus.

         2.      The Agent is authorized and hereby agrees to act as follows:

                 (a)      to address, and deliver by hand or next day courier,
                          a complete set of the exchange offer documents
                          including, but not limited to, the Prospectus, the
                          LT, the Notice of Guaranteed Delivery and a
                          self-addressed envelope prepared by the Agent, to
                          each person who, prior to the Expiration Date,
                          becomes a registered holder of Old Debentures
                          promptly after such person becomes a registered
                          holder of Old Debentures;
<PAGE>   2





                 (b)      to receive all tenders of Old Debentures made
                          pursuant to the Exchange Offer and stamp the LT with
                          the day, month and approximate time of receipt;

                 (c)      to examine each LT and Old Debenture received to
                          determine that all requirements necessary to
                          constitute a valid tender have been met.  The Agent
                          shall be entitled to rely on the electronic messages
                          sent by the Depository Trust Company ("DTC") in
                          accordance with DTC's Automated Tender Offer Program
                          to the Agent's account at DTC from the DTC
                          participants listed on the DTC position listing
                          provided to the Agent;

                 (d)      to take such actions necessary and appropriate to
                          correct any irregularity or deficiency associated
                          with any tender not in proper order;

                 (e)      to follow instructions give by Randall E. Curran,
                          James H. Tate and Stephanie N. Josephson, each an
                          officer of the Company, or such other person or
                          persons as may be designated by the Company in
                          writing, with respect to the waiver of any
                          irregularities or deficiencies associated with any
                          tender;

                 (f)      to hold all valid tenders subject to further
                          instructions from Randall E. Curran, James H. Tate or
                          Stephanie N. Josephson, each an officer of the
                          Company;

                 (g)      to render a written report to the Company and Weil,
                          Gotshal & Manges LLP, in the form of Exhibit A
                          attached hereto, on each business day during the
                          Exchange Offer and promptly confirm, by telephone,
                          the information contained therein to Stephanie N.
                          Josephson at the Company at (314) 746-2139 and
                          Michael S. Colvin at Weil, Gotshal & Manges LLP at
                          (214) 746-7790.

                 (h)      to follow and act upon any written amendments,
                          modifications or supplements to these instructions,
                          any of which may be given to the Agent by Randall E.
                          Curran, James H. Tate or Stephanie N. Josephson, each
                          an officer of the Company, or such other person or
                          persons as they shall designate in writing;

                 (i)      to return to the presenters, in accordance with the
                          provisions of the LT, any Old Debentures that were
                          not received in proper order and as to which the
                          irregularities or deficiencies were not cured or
                          waived;

                 (j)      in the event the Exchange Offer is consummated, to
                          deliver authenticated Exchange Debentures to
                          tendering Holders, in accordance with the
                          instructions of such Holders specified in the
                          respective LT's, as soon as practicable after receipt
                          thereof;
<PAGE>   3





                 (k)      to determine that all endorsements, guarantees,
                          signatures, authorities, stock transfer taxes (if
                          any) and such other requirements are fulfilled in
                          connection with any request for issuance of the
                          Exchange Debentures in a name other than that of the
                          registered owner of the Old Debentures;

                 (l)      to deliver to, or upon the order of, the Company all
                          Old Debentures received under the Exchange Offer,
                          together with any related assignment forms and other
                          documents; and

                 (m)      subject to other terms and conditions set forth in
                          this Agreement to take all other actions reasonable
                          and necessary in the good faith judgment of the
                          Agent, to effect the foregoing matters.

         3.      The Agent shall:

                 (a)      have no duties or obligations other than those
                          specifically set forth herein;

                 (b)      not be required to refer to any documents for the
                          performance of its obligations hereunder other than
                          this Agreement, the LT and the documents required to
                          be submitted with the LT; other than such documents,
                          the Agent will not be responsible or liable for any
                          terms, directions or information in the Prospectus or
                          any other document or agreement unless the Agent
                          specifically agrees thereto in writing;

                 (c)      not be required to act on the directions of any
                          person, including the persons named above, unless the
                          Company provides a corporate resolution to the Agent
                          or other evidence satisfactory to the Agent of the
                          authority of such person;

                 (d)      not be required to and shall make no representations
                          and have no responsibilities as to the validity,
                          accuracy, value or genuineness of (i) the Exchange
                          Offer, (ii) any certificates, LT's or documents
                          prepared by the Company in connection with the
                          Exchange Offer or (iii) any signatures or
                          endorsements, other than its own;

                 (e)      not be obligated to take any legal action hereunder
                          that might, in its judgment, involve any expense or
                          liability, unless it has been furnished with
                          reasonable indemnity by the Company;





                                       3


<PAGE>   4





                 (f)      be able to rely on and shall be protected in acting
                          on the written or oral instructions with respect to
                          any matter relating to its actions as Agent
                          specifically covered by this Agreement, of any
                          officer of the Company authorized to give
                          instructions under paragraphs 2(e) or 2(h) above;

                 (g)      be able to rely on and shall be protected in acting
                          upon any certificate, instrument, opinion, notice,
                          letter, telegram or any other document or security
                          delivered to it and believed by it reasonably and in
                          good faith to be genuine and to have been signed by
                          the proper party or parties;

                 (h)      not be responsible for or liable in any respect on
                          account of the identity, authority or rights of any
                          person executing or delivering or purporting to
                          execute or deliver any document or property under
                          this Agreement and shall have no responsibility with
                          respect to the use or application of any property
                          delivered by it pursuant to the provisions hereof;

                 (i)      be able to consult with counsel satisfactory to it
                          (including counsel for the Company or staff counsel
                          of the Agent) and the advice or opinion of such
                          counsel shall be full and complete authorization and
                          protection in respect of any action taken, suffered
                          or omitted by it hereunder in good faith and in
                          accordance with advice or opinion of such counsel;

                 (j)      not be called on at any time to advise, and shall not
                          advise, any person delivering an LT pursuant to the
                          Exchange Offer as to the value of the consideration
                          to be received;

                 (k)      not be liable for anything which it may do or refrain
                          from doing in connection with this Agreement except
                          for its own gross negligence, willful misconduct or
                          bad faith;

                 (l)      not be bound by any notice or demand, or any waiver
                          or modification of this Agreement or any of the terms
                          hereof, unless evidenced by a writing delivered to
                          the Agent signed by the proper authority or
                          authorities and, if the Agent's duties or rights are
                          affected, unless the Agent shall give its prior
                          written consent thereto;

                 (m)      have no duty to enforce any obligation of any person
                          to make delivery, or to direct or cause any delivery
                          to be made, or to enforce any obligation of any
                          person to perform any other act; and





                                       4


<PAGE>   5





                 (n)      have the right to assume, in the absence of written
                          notice to the contrary from the proper person or
                          persons, that a fact or an event by reason of which
                          an action would or might be taken by the Agent does
                          not exist or has not occurred without incurring
                          liability for any action taken or omitted, or any
                          action suffered by the Agent to be taken or omitted,
                          in good faith or in the exercise of the Agent's best
                          judgment, in reliance upon such assumption.

         4.      The Agent shall be entitled to compensation as set forth in
Exhibit B attached hereto.

         5.      The Company covenants and agrees to reimburse the Agent for,
indemnify it against, and hold it harmless from any and all reasonable costs
and expenses (including reasonable fees and expenses of counsel and allocated
cost of staff counsel) that may be paid or incurred or suffered by it or to
which it may become subject without gross negligence, willful misconduct or bad
faith on its part by reason of or as a result of its compliance with the
instructions set forth herein or with any additional or supplemental written or
oral instructions delivered to it pursuant hereto, or which may arise out of or
in connection with the administration and performance of its duties under this
Agreement.  The Company agrees to promptly notify the Agent of any extension of
the Expiration Date.

         6.      This Agreement shall be construed and enforced in accordance
with the laws of the State of New York and shall inure to the benefit of, and
the obligations created hereby shall be binding upon, the successors and
assigns of the parties thereto.  The parties agree to submit to the exclusive
jurisdiction of the federal or state courts located in the State of New York,
New York County.

         7.      Unless otherwise expressly provided herein, all notices,
requests, demands and other communications hereunder shall be in writing, shall
be delivered by hand, facsimile or by first class mail, postage prepaid, shall
be deemed given when received and shall be addressed to the Agent and the
Company at the respective addresses listed below or to such other addresses as
they shall designate from time to time in writing, forwarded in like manner.

           If to the Agent, to:      IBJ Schroder Bank & Trust Company
                                     One State Street
                                     New York, NY  10004
                                     Attention:  Reorganization Operations Dept.
                                     Telephone: (212) 858-2103
                                     Facsimile:  (212) 858-2611





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<PAGE>   6





           with copies to:          IBJ Schroder Bank & Trust Company
                                    One State Street
                                    New York, NY  10004
                                    Attention:  Corporate Finance Trust Services
                                    Telephone: (212) 858-2176
                                    Facsimile:  (212) 858-2952

           To the Company:          Thermadyne Holdings Corporation
                                    101 South Hanley Road, Suite 300
                                    St. Louis, Missouri 63105
                                    Attention:  Stephanie N. Josephson
                                    Telephone: (314) 746-2139
                                    Facsimile:  (314) 746-2327

          with copies to:           Weil, Gotshal & Manges LLP
                                    100 Crescent Court, Suite 1300
                                    Dallas, Texas  75201-6950
                                    Attention:  R. Scott Cohen
                                    Telephone: (214) 746-7738
                                    Facsimile:  (214) 746-7777





            [THE REMAINDER OF THIS PAGE IS INTENTIONALLY LEFT BLANK]





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<PAGE>   7





                 IN WITNESS WHEREOF, the parties hereto have caused this
Agreement to be executed on their behalf by their officers thereunto duly
authorized, all as of the day and year first written above.

                                         IBJ SCHRODER BANK & TRUST COMPANY



                                         By:                                   
                                            -----------------------------------
                                         Name:                                 
                                              ---------------------------------
                                         Title:                                
                                               --------------------------------



                                         THERMADYNE HOLDINGS CORPORATION



                                         By:                                   
                                            -----------------------------------
                                         Name:                                 
                                              ---------------------------------
                                         Title:
                                               --------------------------------





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<PAGE>   8





                                   EXHIBIT A

                                 SAMPLE REPORT

<TABLE>
<S>                                                                          <C>
                                                                             Date:                                       
                                                                                  ---------------------------------------
                                                                             Report Number:                              
                                                                                           ------------------------------
                                                                             As of Date:                                 
                                                                                        ---------------------------------


Ladies & Gentlemen:

As Exchange Agent for the Exchange Offer dated ____________, 1998, we hereby render the following report:

Principal Amount previously received:                                                                                    
                                                                    -----------------------------------------------------

Principal Amount received today:                                                                                         
                                                                    -----------------------------------------------------

Principal Amount received against Guaranteed Deliveries:                                                                 
                                                                    -----------------------------------------------------

Principal Amount withdrawn today:                                                                                        
                                                                    -----------------------------------------------------

TOTAL PRINCIPAL AMOUNT RECEIVED TO DATE:                                                                                 
                                                            -------------------------------------------------------------

RECAP OF PRINCIPAL AMOUNT REPRESENTED BY GUARANTEES

Guarantees previously outstanding:                                                                                       
                                                                    -----------------------------------------------------

Guarantees received today:                                                                                               
                                                                    -----------------------------------------------------

Guarantees settled today:                                                                                                
                                                                    -----------------------------------------------------

Guarantees withdrawn today:                                                                                              
                                                                    -----------------------------------------------------

Guarantees outstanding:                                                                                                  
                                                                    -----------------------------------------------------

TOTAL PRINCIPAL AMOUNT AND GUARANTEES OUTSTANDING:                                                                       
                                                                    =====================================================


                                                                    Very truly yours,




                                                                    Reorganization Operations Dept.
</TABLE>





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<PAGE>   9





                                   EXHIBIT B

                           AGENT'S FEES AND EXPENSES





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