Experimental efforts on topological superconductivity (TSC) have primarily focused on the detection of Majorana boundary modes, while the bulk properties of TSC, particularly in two dimensions, remain relatively underexplored. In this work, we theoretically propose a distinctive signature in the spectral function away from the boundaries, capable of detecting two-dimensional (2D) chiral p-wave TSC induced in a Rashba spin-orbit-coupled heterostructure. This signature can be probed experimentally through angle-resolved photoemission spectroscopy or momentum- and energy-resolved tunneling spectroscopy under a weak magnetic field B. We show that within the topological phase, the spectral intensity of the lowest superconducting band at small momenta k∼0 brightens (darkens) linearly with increasing B, whereas it darkens (brightens) in the trivial phase when the Rashba system is electron (hole) doped. This sharp contrast arises from the phase-space Berry curvature (BC) of Bogoliubov quasiparticles, a novel quantum geometric property that generalizes the conventional momentum-space BC. The effect of this phase-space BC can also be detected by the differential conductance away from the boundaries. Our falsifiable prediction provides an experimental avenue for detecting Rashba-induced chiral p-wave TSC without relying on Majorana mode detection, addressing a key challenge in the realization of 2D TSC.