  We prove that the implicit time Euler scheme coupled with finite elements
space discretization for the 2D Navier-Stokes equations on the torus subject to
a random perturbation converges in $L^2(\Omega)$, and describe the rate of
convergence for an $H^1$-valued initial condition. This refines previous
results which only established the convergence in probability of these
numerical approximations. Using exponential moment estimates of the solution of
the stochastic Navier-Stokes equations and convergence of a localized scheme,
we can prove strong convergence of this space-time approximation. The speed of
the $L^2(\Omega)$-convergence depends on the diffusion coefficient and on the
viscosity parameter. In case of Scott-Vogelius mixed elements and for an
additive noise, the convergence is polynomial.
