A dual-functional ultraviolet (UV) photodetector with a large UV-to-visible rejection ratio is presented, in which interdigitated finger-type two-dimensional graphene electrodes are introduced to an AlGaN/GaN heterostructure. Two photocurrent generation mechanisms of photovoltaic and photoconductive dominances coexist in the device. The dominance of the mechanisms changes with the induced bias voltage. Below a threshold voltage, the device showed fairly low responsivities but fast response times, as well as a constant photocurrent against the induced bias. However, the opposite characteristics appeared with high bias voltage. Specifically, above the threshold voltage, the device showed high responsivities with additional gain, but slow rise and recovery times. For instance, the responsivity of 10.9 A/W was observed with the gain of 760 at the induced bias voltage of 5 V. This unique multifunctionality enabled by the combination of an AlGaN/GaN heterostructure with graphene electrodes facilitates the development of a single device that can achieve multiple purposes of photodetection.