Hybrid graphene photoconductor/phototransistor has achieved giant photoresponsivity, but its response speed dramatically degrades as the expense due to the long lifetime of trapped interfacial carriers. In this work, by intercalating a large-area atomically thin MoS2 film into a hybrid graphene photoconductor, we have developed a prototype tunneling photoconductor, which exhibits a record-fast response (rising time ~17 ns) and a high responsivity (~$3times10^4$ A/W at 635 nm and 16.8 nW illumination) across the broad spectral range. We demonstrate that the photo-excited carriers generated in silicon are transferred into graphene through a tunneling process rather than carrier drift. The atomically thin MoS2 film not only serves as tunneling layer but also passivates surface states, which in combination delivers a superior response speed (~3 order of magnitude improved than a device without MoS2 layer), while the responsivity remains high. This intriguing tunneling photoconductor integrates both fast response and high responsivity and thus has significant potential in practical applications of optoelectronic devices.