The non trivial zeros respective poles of $\zeta$ lie outside $ Re s > 1$ follows from the absolute convergence of the Euler product in this region and does not need any entire function theory, not even holomorphicity. That the Riemann Zeta function is holomorphic, actually follows from the absoulute locally uniform convergence of the product. The functional equation follows from the Poisson summation formula (=harmonic analysis). Having the functional equation and the nonvanishing in $Re s>1$ established, we can apply a contour integration (complex analysis), but we need certain boundedness conditions in the critical region $ 0 \leq Re s \leq 1$. This needs the Hadamard three lines principle or the Phragmen Lindeloeff principle (this is not entire function theory, but this only a complex analysis argument), then the explicite formula follows.

If you interested in the merely existence of nontrivial zeros without using meromorphicity of $\zeta$, you might want to try to deduce this from knowledge over the primes and by inserting an appropiete chosen test function in the explicit formula. It might be helpful to use the analogy with the Selberg trace formula and the derivation of the Weyl law. Werner Mueller has some nice expository articles here, which show how to count the eigenvalues of the Laplace Beltrami or the zeros of the Selberg Zeta function with choosing a set of suitable test function and applying the Selberg trace formula. 

One proof using only the languages of the adeles, harmonic analysis and no entire function theory explicitely at all was given by Ralf Meyer:
http://arxiv.org/pdf/math/0311468v3

However the Fourier transform of compact supported function is always entire, so you basically just hide the complex analysis arguments.