When computing scattering amplitudes in dimensional regularization, one frequently encounters contributions whose integrands vanish in strictly four dimensions. While these "evanescent" integrals can be handled with dimensional shift identities at one-loop, a similar treatment at the next perturbative order is insufficient. In this talk, we introduce a novel systematic method to compute evanescent contributions. By employing the local subtraction method of Anastasiou and Sterman we show that evanescent Feynman integrals are controlled by regions of loop-momentum space associated to ultra-violet, soft or collinear divergences. These integrals are then reduced to either products of one-loop integrals or one-fold integrals thereof. Starting from known integrands, we use this technique to easily recompute the leading-color two-loop four- and five-gluon QCD amplitudes in the all-plus helicity configuration. Remarkably, we find that the finite remainder is given by contributions arising from only ultra-violet regions of momentum space, and that the collinear contributions cancel in a highly non-trivial way.