In this work we prove an analogue, for partial differential equations on the space of probability measures, of the classical vanishing viscosity result known for equations on the Euclidean space. Our result allows in particular to show that the value function arising in various problems of classical mechanics and games can be obtained as the limiting case of second order PDEs. The method of proof builds on stochastic analysis arguments and allows for instance to prove a Freindlin-Wentzell large deviation theorem for McKean-Vlasov equations.