The scaling of the mobility of two-dimensional Langevin dynamics in a periodicpotential as the friction vanishes is not well understood for nonseparable potentials. Theoreticalresults are lacking, and numerical calculation of the mobility in the underdamped regime is chal-lenging because the computational cost of standard Monte Carlo methods is inversely proportionalto the friction coefficient, while deterministic methods are ill-conditioned. In this work, we proposea new variance-reduction method based on control variates for efficiently estimating the mobility ofLangevin-type dynamics. We provide bounds on the bias and variance of the proposed estimatorand illustrate its efficacy through numerical experiments, first in simple one-dimensional settingsand then for two-dimensional Langevin dynamics. Our results corroborate prior numerical evidencethat the mobility scales as\gamma - \sigma , with 0<\sigma \leqslant 1, in the low friction regime for a simple nonseparablepotential.