Chiral active matter is predicted to exhibit odd elasticity, with nontraditional elastic response arising from a combination of chirality, being out of equilibrium, and the presence of nonreciprocal interactions. One of the resulting phenomena is the possible occurrence of odd elastic waves in overdamped systems, although its experimental realization still remains elusive. Here we show that in overdamped active systems, noise is required to generate persistent elastic waves. In the chiral crystalline phase of active matter, such as that found recently in populations of swimming starfish embryos, the noise arises from the self-driving of active particles and their mutual collisions, a key factor that has been missing in previous studies. We identify the criterion for the occurrence of noise-driven odd elastic waves and construct the corresponding phase diagram, which is also applicable to general chiral active crystals. Our results can be used to predict the experimental conditions for achieving a transition to self-sustained elastic waves in overdamped active systems.