I think that you need some extra condition for this result to hold. Suppose for example that

Then is analytic (as a function of z) for each fixed But it is not (Riemann) integrable as a function of t, except at z=0.

If you have some sufficiently strong additional condition controlling the integrability of f as a function of t, then you could hope to prove the result by approximating the integral by Riemann sums, and using the fact (corollary of Morera's theorem) that a uniform limit of analytic functions is analytic.