Recent observations have shown several supernova remnants (SNRs) have overionized plasmas, those where ions are stripped of more electrons than they would be if in equilibrium with the electron temperature. Rapid electron cooling is necessary to produce this situation, yet the physical origin of that cooling remains uncertain. To assess the cooling scenario responsible for overionization, in this paper, we identify and map the overionized plasma in the Galactic SNR W49B based on a 220 ks Chandra Advanced CCD Imaging Spectrometer (ACIS) observation. We performed a spatially-resolved spectroscopic analysis, measuring the electron temperature by modeling the continuum and comparing it to the temperature given by the flux ratio of the He-like and H-like lines of sulfur and of argon. Using these results, we find that W49B is overionized in the west, with a gradient of increasing overionization from east to west. As the ejecta expansion is impeded by molecular material in the east but not in the west, our overionization maps suggest the dominant cooling mechanism is adiabatic expansion of the hot plasma.