<PAGE>   1
                       SECURITIES AND EXCHANGE COMMISSION

                             WASHINGTON, D. C. 20549

                          -------------------------

                                    FORM T-1

                            STATEMENT OF ELIGIBILITY
                   UNDER THE TRUST INDENTURE ACT OF 1939 OF A
                    CORPORATION DESIGNATED TO ACT AS TRUSTEE

                      CHECK IF AN APPLICATION TO DETERMINE
                      ELIGIBILITY OF A TRUSTEE PURSUANT TO
                                SECTION 305(b)(2)_

                          -------------------------

                        IBJ SCHRODER BANK & TRUST COMPANY
               (Exact name of trustee as specified in its charter)

         New York                                              13-5375195
(Jurisdiction of incorporation                            (I.R.S. Employer
or organization if not a U.S. national bank)              Identification No.)


One State Street, New York, New York                         10004
(Address of principal executive offices)                   (Zip code)

                        IBJ SCHRODER BANK & TRUST COMPANY
                                One State Street
                            New York, New York 10004
                                 (212) 858-2000
            (Name, address and telephone number of agent for service)

                         THERMADYNE HOLDINGS CORPORATION
               (Exact name of obligor as specified in its charter)

         Delaware                                           74-2482571
(State or other jurisdiction of                          (I.R.S. Employer
incorporation or organization)                           Identification No.)

     101 South Hanley Road, Suite 300
         St. Louis, Missouri                                 63105
(Address of principal executive offices)                  (Zip code)

                          -------------------------

                   12 1/2% SENIOR DISCOUNT DEBENTURES DUE 2008
                         (Title of indenture securities)


                   ----------------------------------------

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Item 1. General information

        Furnish the following information as to the trustee:

        (a)     Name and address of each examining or supervising authority to
                which it is subject.

                New York State Banking Department Two Rector Street, New York,
                New York

                Federal Deposit Insurance Corporation, Washington, D.C.

                Federal Reserve Bank of New York Second District 33 Liberty
                Street, New York, New York

        (b)     Whether it is authorized to exercise corporate trust powers.

                Yes

Item 2. Affiliations with the Obligor.

                If the obligor is an affiliate of the trustee, describe each
                such affiliation.

                The obligor is not an affiliate of the trustee.

Item 4. Trusteeships under other indentures.

                If the trustee is a trustee under another indenture under which
                any other securities, or certificates of interest or
                participation in any other securities of the obligor are
                outstanding, furnish the following information:

        (a)     Title of the securities outstanding under each such other
                indenture.

                None    

        (b)     A brief statement of the facts relied upon as a basis for the
                claim that no conflicting interest within the meaning of Section
                (310)(b)(1) of the Act arises as a result of the trusteeship
                under any such other indenture, including a statement as to how
                the indenture securities will rank as compared with the
                securities issued under such other indentures.

                Not applicable


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Item 13. Defaults by the Obligor.

        (a)     State whether there is or has been a default with respect to the
                securities under this indenture. Explain the nature of any such
                default.

                None

        (b)     If the trustee is a trustee under another indenture under which
                any other securities, or certificates of interest or
                participation in any other securities, of the obligor are
                outstanding, or is trustee for more than one outstanding series
                of securities under the indenture, state whether there has been
                a default under any such indenture or series, identify the
                indenture or series affected, and explain the nature of any such
                default.

                Not applicable



<PAGE>   4



Item 16. LIST OF EXHIBITS.

                List below all exhibits filed as part of this statement of
                eligibility.

                *1.     A copy of the Charter of IBJ Schroder Bank & Trust
                        Company as amended to date. (See Exhibit 1A to Form T-1,
                        Securities and Exchange Commission File No. 22-18460).

                *2.     A copy of the Certificate of Authority of the trustee to
                        Commence Business (Included in Exhibit 1 above).

                *3.     A copy of the Authorization of the trustee to exercise
                        corporate trust powers, as amended to date (See Exhibit
                        4 to Form T-1, Securities and Exchange Commission File
                        No. 22-19146).

                *4.     A copy of the existing By-Laws of the trustee, as
                        amended to date (See Exhibit 4 to Form T-1, Securities
                        and Exchange Commission File No. 22-19146).

                 5.     Not Applicable

                 6.     The consent of United States institutional trustee
                        required by Section 321(b) of the Act.

                 7.     A copy of the latest report of condition of the trustee
                        published pursuant to law or the requirements of its
                        supervising or examining authority.

*       The Exhibits thus designated are incorporated herein by reference as
        exhibits hereto. Following the description of such Exhibits is a
        reference to the copy of the Exhibit heretofore filed with the
        Securities and Exchange Commission, to which there have been no
        amendments or changes.


<PAGE>   5

                                      NOTE

In answering any item in this Statement of Eligibility which relates to matters
peculiarly within the knowledge of the obligor and its directors or officers,
the trustee has relied upon information furnished to it by the obligor.

Inasmuch as this Form T-1 is filed prior to the ascertainment by the trustee of
all facts on which to base responsive answers to Item 2, the answer to said Item
are based on incomplete information.

Item 2, may, however, be considered as correct unless amended by an amendment to
this Form T-1.


<PAGE>   6


                                    SIGNATURE

Pursuant to the requirements of the Trust Indenture Act of 1939, the trustee,
IBJ Schroder Bank & Trust Company, a corporation organized and existing under
the laws of the State of New York, has duly caused this statement of eligibility
to be signed on its behalf by the undersigned, thereunto duly authorized, all in
the City of New York, and State of New York, on the 13th day of July, 1998.



                                    IBJ SCHRODER BANK & TRUST COMPANY



                                    By: /s/ Stephen J. Giurlando
                                        ----------------------------
                                            Stephen J. Giurlando
                                            Assistant Vice President


<PAGE>   7


                                    EXHIBIT 6

                               CONSENT OF TRUSTEE


Pursuant to the requirements of Section 321(b) of the Trust Indenture Act of
1939, as amended, in connection with the issuance by Thermadyne Holdings
Corporation of its 12 1/2% Senior Discount Debentures due 2008, we hereby
consent that reports of examinations by Federal, State, Territorial, or District
authorities may be furnished by such authorities to the Securities and Exchange
Commission upon request therefor.


                                    IBJ SCHRODER BANK & TRUST COMPANY



                                    By: /s/ Stephen J. Giurlando
                                       ---------------------------------
                                            Stephen J. Giurlando
                                            Assistant Vice President


Dated: July 13, 1998


<PAGE>   8

                                    EXHIBIT 7


                       CONSOLIDATED REPORT OF CONDITION OF
                        IBJ SCHRODER BANK & TRUST COMPANY
                              OF NEW YORK, NEW YORK
                      AND FOREIGN AND DOMESTIC SUBSIDIARIES


                           REPORT AS OF MARCH 31, 1998



<TABLE>
<CAPTION>
                                                                                                      DOLLAR AMOUNTS
                                                                                                       IN THOUSANDS
                                                                                                       ------------
                                     ASSETS
                                     ------
<CAPTION>
<S>  <C>                                                                                             <C>        
1.   Cash and balance due from depository institutions:
     a.  Noninterest-bearing balances and currency and coin   .......................................$    29,353
     b.  Interest-bearing balances...................................................................$    15,329

2.   Securities:
     a.  Held-to-maturity securities.................................................................$   186,942
     b.  Available-for-sale securities...............................................................$   102,403

3.   Federal funds sold and securities purchased under agreements to resell in
     domestic offices of the bank and of its Edge and Agreement subsidiaries and
     in IBFs:

     Federal Funds sold and Securities purchased under agreements to resell..........................$   176,231

4.   Loans and lease financing receivables:
     a.  Loans and leases, net of unearned income......................................$ 1,673,749
     b.  LESS: Allowance for loan and lease losses.....................................$    63,611
     c.  LESS: Allocated transfer risk reserve.........................................$        -0-
     d.  Loans and leases, net of unearned income, allowance, and reserve............................$ 1,610,138

5.   Trading assets held in trading accounts.........................................................$       584

6.   Premises and fixed assets (including capitalized leases)........................................$     2,575

7.   Other real estate owned.........................................................................$       819

8.   Investments in unconsolidated subsidiaries and associated companies.............................$        -0-

9.   Customers' liability to this bank on acceptances outstanding....................................$       503

10.  Intangible assets...............................................................................$        -0-

11.  Other assets....................................................................................$    61,923

12.  TOTAL ASSETS....................................................................................$ 2,186,800
</TABLE>

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<TABLE>
<CAPTION>
                                   LIABILITIES
                                   -----------


<S>  <C>           
13.  Deposits:                                                                   <C>               <C>
     a.  In domestic offices.......................................................................$      659,051

     (1) Noninterest-bearing ....................................................$      288,134
     (2) Interest-bearing . . . . . . . . . . . . . . . . . . . . . . . . . . . .$      370,917

     b.  In foreign offices, Edge and Agreement subsidiaries, and IBFs.............................$    1,141,113

     (1) Noninterest-bearing . . . . . . . . . . . . . . . . . . . . . . . . . . $      19,428
     (2) Interest-bearing . . . . . . . . . . . . . . . . . . . . . . . . . . . .$   1,121,685

14.  Federal funds purchased and securities sold under agreements to repurchase
     in domestic offices of the bank and of its Edge and Agreement subsidiaries,
     and in IBFs:

     Federal Funds purchased and Securities sold under agreements to repurchase....................$           -0-

15.  a.  Demand notes issued to the U.S. Treasury..................................................$        5,000

     b.  Trading Liabilities.......................................................................$          344

16. Other borrowed money:
     a.  With a remaining maturity of one year or less.............................................$       61,953
     b.  With a remaining maturity of more than one year...........................................$        1,763
     c.  With a remaining maturity of more than three years........................................$        2,242

17. Not applicable.

18.  Bank's liability on acceptances executed and outstanding......................................$          503

19.  Subordinated notes and debentures.............................................................$           -0-

20.  Other liabilities.............................................................................$       70,344

21.  TOTAL LIABILITIES.............................................................................$    1,942,313

22.  Limited-life preferred stock and related surplus..............................................$          N/A


                                 EQUITY CAPITAL


23.  Perpetual preferred stock and related surplus.................................................$           -0-

24.  Common stock..................................................................................$       29,649

25.  Surplus (exclude all surplus related to preferred stock)......................................$      217,008

26.  a.  Undivided profits and capital reserves....................................................$       (2,291)

     b.  Net unrealized gains (losses) on available-for-sale securities............................$          121

27.  Cumulative foreign currency translation adjustments...........................................$           -0-

28.  TOTAL EQUITY CAPITAL..........................................................................$      244,487

29.  TOTAL LIABILITIES AND EQUITY CAPITAL..........................................................$    2,186,800
</TABLE>