The claim in the title is proved on pp.19-20 of [Topological rigidity for non-aspherical manifolds](https://arxiv.org/pdf/math/0509238.pdf)
by M. Kreck and W. Lueck. Is there an earlier (classical) reference?



Where was it first shown that every homotopy self-equivalence of $S^1\times S^2$ is homotopic to a homeomorphism?
My guess is that the oldest reference might be Pontryagin's 1941 paper on the homotopy classification of maps from a 3-dimensional complex to the 2-sphere, the English version of which is in Recueil Mathématique 51 pp. 331-359. The application to homotopy equivalences of $S^1\times S^2$ is in an example on page 356 at the end of Section 4 of the paper.


