Well, a simple counter example is $$A(s)=\sum_{n=1}^\infty a_n n^{-s}= e^{\eta(s)},$$
where 
$$\eta(s)=\sum_{n=1}^\infty (-1)^{n-1} n^{-s}=(1-2^{1-s})\zeta(s).$$
This Dirichlet series is obviously meromorphic since it is in fact entire and it is also absolutely convergent on some half plane Re$(s)>c$. This entire function is not of finite order by the functional equation of the Riemann zeta-function and Stirling's formula.