Correlated charge inhomogeneity breaks the electron-hole symmetry in two-dimensional (2D) bilayer heterostructures which is responsible for nonzero drag appearing at the charge neutrality point. Here we report Coulomb drag in novel drag systems consisting of a two-dimensional graphene and a one-dimensional (1D) InAs nanowire (NW) heterostructure exhibiting distinct results from 2D-2D heterostructures. For monolayer graphene (MLG)-NW heterostructures, we observe an unconventional drag resistance peak near the Dirac point due to the correlated interlayer charge puddles. The drag signal decreases monotonically with temperature (∼T^{-2}) and with the carrier density of NW (∼n_{N}^{-4}), but increases rapidly with magnetic field (∼B^{2}). These anomalous responses, together with the mismatched thermal conductivities of graphene and NWs, establish the energy drag as the responsible mechanism of Coulomb drag in MLG-NW devices. In contrast, for bilayer graphene (BLG)-NW devices the drag resistance reverses sign across the Dirac point and the magnitude of the drag signal decreases with the carrier density of the NW (∼n_{N}^{-1.5}), consistent with the momentum drag but remains almost constant with magnetic field and temperature. This deviation from the expected T^{2} arises due to the shift of the drag maximum on graphene carrier density. We also show that the Onsager reciprocity relation is observed for the BLG-NW devices but not for the MLG-NW devices. These Coulomb drag measurements in dimensionally mismatched (2D-1D) systems, hitherto not reported, will pave the future realization of correlated condensate states in novel systems.