In the Mitchell book I.16.8 (page 24) is a Corollary of the $3\times3$ (or nine) Lemma, and this Lemma is true in a exat category (see the Book: H. Shubert "CAtegories" p. 136), then I see that Mitchell dont use addittivity (for demostrate the corillary from the Lemma) then the corollary follow also for exat categories.
Is you dont have the H. Schubert Book I can post a proof here.
PS. for "exat category" Shubert mean: pointed, with $(regular-Epi, regular-Mono)$ factorization.