Although several mechanisms can promote cooperative behavior, there is no general consensus about why cooperation survives when the most profitable action for an individual is to defect, especially when the population is well mixed. Here we show that when a replicator such as evolutionary game dynamics takes place on interdependent networks, cooperative behavior is fixed on the system. Remarkably, we analytically and numerically show that this is even the case for well-mixed populations. Our results open the path to mechanisms able to sustain cooperation and can provide hints for controlling its rise and fall in a variety of biological and social systems.