In simulations of viscously evolving accretion disks, the inner boundary condition is particularly important. If treated incorrectly, it induces incorrect behavior very quickly, because the viscous time is shortest near the inner boundary. Recent work has determined the correct inner boundary in Eulerian simulations. But in quasi-Lagrangian simulations (e.g., SPH, moving mesh, and meshless), where the inner boundary is modeled by removing mass within a finite zone, the inner density profile typically becomes anomalously depleted. Here we show how the boundary condition should be applied in such codes, via a simple modification of the usual approach: When one removes mass, one must speed up the remaining material so that the disk's angular momentum is unchanged. We show with both 1D and 2D moving-mesh (AREPO) simulations that this scheme works as desired in viscously evolving disks. It produces no spurious density depletions and is independent of the mass removal rate, provided that the disk is adequately resolved and that the mass removal rate is not so extreme as to trigger instabilities. This "torque-free" mass removal technique permits the use of quasi-Lagrangian codes to simulate viscously evolving disks, while including a variety of additional effects. As an example, we apply our scheme to a 2D simulation of an accretion disk perturbed by a very massive planet, in which the disk is evolved to viscous steady state.