We propose and fabricate an integrated graphene/glass hybrid photodetector (PD) with high responsivity and broad spectral bandwidth. The glass straight waveguide enables high absorption of the evanescent light of transverse magnetic (TM) mode propagating parallel to the single layer of graphene. It is based on the mechanism of light-induced change in conductance. As a result, a responsivity as high as 0.72 A/W at a low bias voltage of -0.1 V for a wide wavelength range from 1510 to 1630 nm is experimentally obtained. The proposed graphene/glass hybrid PD could find important applications in graphene-based photonic integrated circuits.