This paper studies the vanishing viscosity approximation to mean field games (MFGs) in $\mathbb{R}^d$ with a nonlocal and possibly non-separable Hamiltonian. We prove that the value function converges at a rate of $\mathcal{O}(β)$, where $β^2$ is the diffusivity constant, which matches the classical convergence rate of vanishing viscosity for Hamilton-Jacobi (HJ) equations. The same rate is also obtained for the approximation of the distribution of players as well as for the gradient of the value function. The proof is a combination of probabilistic and analytical arguments by first analyzing the forward-backward stochastic differential equation associated with the MFG, and then applying a general stability result for HJ equations. Applications of our result to $N$-player games, mean field control, and policy iteration for solving MFGs are also presented.