(Just a quick aside: the set of solutions to the equation $\zeta(s) = a$ when $a \neq 0$ are usually called the $a$-points of the Riemann zeta function in the literature, in case you want to look up the state of the art)

As Steve Huntsman mentions in a comment, it is indeed possible to use the [universality of the zeta function][1] to prove that $\zeta(s)$ is surjective with a simple argument based on [Rouché's theorem][2]. 

In particular,  let $U$ be a compact connected subset with a smooth boundary in the critical strip on which universality holds (for simplicity, take $U = \{ z \in \mathbb{C} : |z - 3/4| \leqslant 1/8 \}$), and let $f:U \to \mathbb{C}$ be a function holomorphic on $U$ (i.e. holomorphic in the interior, and continuous on the boundary) with the following properties:

1. $f$ has no zeroes in $U$.

2. There exists a $z \in U$ such that $f(z) = a$. 

3. For all $z \in \partial U$, $f(z) \neq a$.

As an example, for $U$ as above, one could take $f(z) = a + |a| (z-3/4)$.

Now note that 3 implies that there is an $\epsilon$ such that $$0 < \epsilon < \inf_{z \in \partial U} |f(z) - a|.$$

Then, due to 1, universality guarantees that there is a $t \geqslant 0$ such that $$ |\zeta(s+it) - f(s)| < \epsilon$$ for every $s \in U$. In particular, this inequality holds for every $s \in \partial U$. Thus, for every $s \in \partial U$, we have that

$$ |\zeta(s+it) - f(s)| < \epsilon < \inf_{z\in\partial U} |f(z) - a| \leqslant |f(s) - a|. $$

Thus, Rouché's theorem implies that $f(s) - a$ and $f(s) - a + \zeta(s+it) - f(s) = \zeta(s+it) - a$ have the same number of roots in the interior of $U$. Since $f$ has such a root, it follows that $\zeta$ has such a root as well, and we are done.

This argument is robust in a few ways. For one, it applies to other functions which exhibit universality (which is known for fairly large classes of $L$-functions and other zeta-like families of interest). For another, effective versions of Voronin's universality theorem can be used to give a good lower bound for the number of $a$-points in a rectangle within the critical strip. For a third, this argument actually proves that there are simple $a$-points for every $a \neq 0$. 

This argument is probably classical, but I first saw it in [a paper on simple $a$-points by Gonek, Lester and Milinovich][3]. 
 


  [1]: https://en.wikipedia.org/wiki/Zeta_function_universality
  [2]: https://en.wikipedia.org/wiki/Rouch%C3%A9%27s_theorem
  [3]: https://mathscinet.ams.org/mathscinet-getitem?mr=2957199