Compact chip-scale comb sources are of significant interest for many practical applications. Here, we experimentally study the generation of supercontinuum (SC) in an axially varying integrated waveguide. We show that the local tuning of the dispersion enables the continuous blue shift of dispersive waves thanks to their trapping by the strongly compressed pump pulse. This mechanism provides new insight into supercontinuum generation in a dispersion varying integrated waveguide. Pumped close to 2.2 $\mu$m in the femtosecond regime and at a pulse energy of $\sim$ 4 pJ, the output spectrum extends from 1.1 $\mu$m up to 2.76 $\mu$m and shows good coherence properties. Octave-spanning SC is also observed at input energy as low as $\sim$ 0.9 pJ. We show that the supercontinuum is more robust against variations of the input pulse parameters and is also spectrally flatter in our numerically optimized waveguide than in fixed-width waveguides. This research demonstrates the potential of dispersion varying waveguides for coherent SC generation and paves the way for integrated low power applications, such as chip-scale frequency comb generation, precision spectroscopy, optical frequency metrology, and wide-band wavelength division multiplexing in the near-infrared.