Fabrication and optoelectronic applications of graphene based hybrid 2D-1D semiconductor nanostructures have gained tremendous research interest in recent times. Herein, we present a systematic study on the origin and evolution of strong broad band visible and near infrared (NIR) photoluminescence (PL) from vertical ZnO nanorods (NRs) and nanowires (NWs) grown on single layer graphene using both above band gap and sub-band gap optical excitations. High resolution field emission scanning electron microscopy and X-ray diffraction studies are carried out to reveal the morphology and crystalline quality of as-grown and annealed ZnO NRs/NWs on graphene. Room temperature PL studies reveal that besides the UV and visible PL bands, a new near-infrared (NIR) PL emission band appears in the range between 815 nm and 886 nm (1.40–1.52 eV). X-ray photoelectron spectroscopy studies revealed excess oxygen content and unreacted metallic Zn in the as-grown ZnO nanostructures, owing to the low temperature growth by a physical vapor deposition method. Post-growth annealing at 700 °C in the Ar gas ambient results in the enhanced intensity of both visible and NIR PL bands. On the other hand, subsequent high vacuum annealing at 700 °C results in a drastic reduction in the visible PL band and complete suppression of the NIR PL band. PL decay dynamics of green emission in Ar annealed samples show tri-exponential decay on the nanosecond timescale including a very slow decay component (time constant ∼604.5 ns). Based on these results, the NIR PL band comprising two peaks centered at ∼820 nm and ∼860 nm is tentatively assigned to neutral and negatively charged oxygen interstitial (Oi) defects in ZnO, detected experimentally for the first time. The evidence for oxygen induced trap states on the ZnO NW surface is further substantiated by the slow photocurrent response of graphene-ZnO NRs/NWs. These results are important for tunable light emission, photodetection, and other cutting edge applications of graphene-ZnO based 2D-1D hybrid nanostructures.