Exhibit 5.1

[Gibson Dunn & Crutcher LLP Letterhead]

Client: 12412-00007

August 5, 2013

BioCryst Pharmaceuticals, Inc.

4505 Emperor Blvd., Suite 200

Durham, North Carolina 27703

Registration Statement on Form S-3 (File No. 175182)

Ladies and Gentlemen:

We have examined the Registration Statement on Form S-3, File No. 333-175182, as amended (the “Registration Statement”) of BioCryst Pharmaceuticals, Inc., a Delaware corporation (the “Company”), filed with the Securities and Exchange Commission (the “Commission”) pursuant to the Securities Act of 1933, as amended (the “Securities Act”), in connection with the offering by the Company of up to 4,600,000 shares of the Company’s common stock, par value $0.01 per share (the “Shares”). The Shares will be issued pursuant to the Underwriting Agreement, dated July 31, 2013 (the “Underwriting Agreement”), among the Company and the underwriters named therein.

In arriving at the opinion expressed below, we have examined originals, or copies certified or otherwise identified to our satisfaction as being true and complete copies of the originals, of specimen Common Stock certificates and such other documents, corporate records, certificates of officers of the Company and of public officials and other instruments as we have deemed necessary or advisable to enable us to render the opinions set forth below. In our examination, we have assumed without independent investigation the genuineness of all signatures, the legal capacity and competency of all natural persons, the authenticity of all documents submitted to us as originals and the conformity to original documents of all documents submitted to us as copies.

Based upon the foregoing, and subject to the assumptions, exceptions, qualifications and limitations set forth herein, we are of the opinion that the Shares, when issued against payment therefor as set forth in the Registration Statement, will be validly issued, fully paid and non-assessable.

We render no opinion herein as to matters involving the laws of any jurisdiction other than the Delaware General Corporation Law. This opinion is limited to the effect of the current state of the Delaware General Corporation Law and the facts as they presently exist. We

BioCryst Pharmaceuticals, Inc.

August 5, 2013

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assume no obligation to revise or supplement this opinion in the event of future changes in such law or the interpretations thereof or such facts.

We consent to the filing of this opinion as an exhibit to the Registration Statement, and we further consent to the use of our name under the caption “Legal Matters” in the Registration Statement and the prospectus that forms a part thereof. In giving this consent, we do not thereby admit that we are within the category of persons whose consent is required under Section 7 of the Securities Act or the Rules and Regulations of the Commission.