Control of the living cell functions with remarkable reliability despite the stochastic nature of the underlying molecular networks--a property presumably optimized by biological evolution. We ask here to what extent the ability of a stochastic dynamical network to produce reliable dynamics is an evolvable trait. Using an evolutionary algorithm based on a deterministic selection criterion for the reliability of dynamical attractors, we evolve networks of noisy discrete threshold nodes. We find that, starting from any random network, reliability of the attractor landscape can often be achieved with only a few small changes to the network structure. Further, the evolvability of networks toward reliable dynamics while retaining their function is investigated and a high success rate is found.