We present an asymmetrically-gated Graphene Self-Switching Diode (G-SSD) as a new negative differential resistance (NDR) device, and study its transport properties using nonequilibrium Green's function (NEGF) formalism and the Extended Huckel (EH) method. The device exhibits a new NDR mechanism, in which a very small quantum tunnelling current is used to control a much-larger channel conduction current, resulting in a very pronounced NDR effect. This NDR effect occurs at low bias voltages, below 1 V, and results in a very high current peak in the μA range and a high peak-to-valley current ratio (PVCR) of 40. The device has an atomically-thin structure with sub-10 nm dimensions, and does not require any doping or external gating. These results suggest that the device has promising potential in applications such as high frequency oscillators, memory devices, and fast switches.