Silicon nitride (Si3N4) waveguides represent a novel photonic platform that is ideally suited for energy efficient and ultrabroadband nonlinear interactions from the visible to the mid-infrared. Chip-based supercontinuum generation in Si3N4 offers a path towards a fully-integrated and highly compact comb source for sensing and time-and-frequency metrology applications. We demonstrate the first successful frequency comb offset stabilization that utilizes a Si3N4 waveguide for octave-spanning supercontinuum generation and achieve the lowest integrated residual phase noise of any diode-pumped gigahertz laser comb to date. In addition, we perform a direct comparison to a standard silica photonic crystal fiber (PCF) using the same ultrafast solid-state laser oscillator operating at 1 µm. We identify the minimal role of Raman scattering in Si3N4 as a key benefit that allows to overcome the fundamental limitations of silica fibers set by Raman-induced self-frequency shift.