We demonstrate coherent supercontinuum generation (SCG) in a monolithically integrated lithium-niobate waveguide, under the presence of second- and third-order nonlinear effects. We achieve more than two octaves of optical bandwidth in a 0.5-cm-long waveguide with 100-picojoule-level pulses. Dispersion engineering of the waveguide allows for spectral overlap between the SCG and the second harmonic, which enables direct detection of the carrier-envelope offset frequency fCEO using a single waveguide. We measure the fCEO of our femtosecond pump source with a 30-dB signal-to-noise ratio.