<PAGE>   1
                                                                     EXHIBIT 5.1


                   [WEIL, GOTSHAL & MANGES LLP LETTERHEAD]



                                 August 4, 1998


Thermadyne Holdings Corporation
101 South Hanley Road, Suite 300
St. Louis, Missouri  63105


Ladies and Gentlemen:

                 We have acted as counsel to Thermadyne Holdings Corporation, a
Delaware corporation (the "Company"), in connection with the preparation and
filing by the Company of a Registration Statement on Form S-1 (Registration No.
333- 57455) (the "Registration Statement"), filed with the Securities and
Exchange Commission on June 23, 1998 under the Securities Act of 1933, as
amended, relating to $174,000,000 aggregate principal amount at maturity of 12
1/2% Senior Discount Debentures due 2008 (the "New Debentures") of the Company.
The Company proposes to offer (the "Exchange Offer"), upon the terms set forth
in the Prospectus contained in the Registration Statement, to exchange $1,000
principal amount at maturity of New Debentures for each $1,000 principal amount
at maturity of issued and outstanding 12 1/2% Senior Discount Debentures due
2008 (the "Old Debentures") of the Company.  The New Debentures will be issued
under the Indenture, dated May 22, 1998, by and between the Company and IBJ
Schroder Bank & Trust Company (the "Trustee") (as amended or supplemented to
the date hereof, the "Indenture").

                 In so acting, we have examined originals or copies, certified
or otherwise identified to our satisfaction, of the Indenture, the form of New
Debentures set forth in the Indenture and such corporate records, agreements,
documents and other instruments, and such certificates or comparable documents
of public officials and of officers and representatives of the Company, and
have made such inquiries of such officers and representatives as we have deemed
relevant and necessary as a basis for the opinion hereinafter set forth.
<PAGE>   2


Thermadyne Holdings Corporation
August 4, 1998
Page 2




                 In such examination, we have assumed the genuineness of all
signatures, the legal capacity of natural persons, the authenticity of all
documents submitted to us as originals, the conformity to original documents of
all documents submitted to us as certified, conformed or photostatic copies and
the authenticity of the originals of such latter documents.  As to all
questions of fact material to this opinion that have not been independently
established, we have relied upon certificates or comparable documents of
officers and representatives of the Company.

                 Based on the foregoing, and subject to the qualifications
stated herein, we are of the opinion that, assuming that the Indenture has been
duly authorized, executed and delivered by the Trustee, when (i) the New
Debentures issuable upon consummation of the Exchange Offer have been duly
executed by the Company and authenticated by the Trustee in accordance with the
terms of the Indenture and (ii) the New Debentures issuable upon consummation
of the Exchange Offer have been duly delivered against receipt of Old
Debentures surrendered in exchange therefor, the New Debentures issuable upon
consummation of the Exchange Offer will constitute the legal, valid and binding
obligations of the Company, enforceable against it in accordance with their
terms, subject to applicable bankruptcy, insolvency, fraudulent conveyance,
reorganization, moratorium and similar laws affecting creditors' rights and
remedies generally, and subject, as to enforceability, to general principles of
equity, including principles of commercial reasonableness, good faith and fair
dealing (regardless of whether enforcement is sought in a proceeding at law or
in equity).

                 The opinion expressed herein is limited to the laws of the
State of New York, the corporate laws of the State of Delaware and the federal
laws of the United States, and we express no opinion as to the effect on the
matters covered by this letter of the laws of any other jurisdiction.

                 The opinion expressed herein is rendered solely for your
benefit in connection with the transactions described herein.  The opinion may
not be used or relied upon by any other person, nor may this letter or any
copies thereof be furnished to a third party, filed with a governmental agency,
quoted, cited or otherwise referred to without our prior written consent.





<PAGE>   3


Thermadyne Holdings Corporation
August 4, 1998
Page 3




                 We hereby consent to the filing of this opinion as an exhibit
to the Registration Statement and the reference to this firm under the caption
"Legal Matters" in the Prospectus forming a part of the Registration Statement.

                                        Very truly yours,

                                        /s/ Weil, Gotshal & Manges LLP