Energy-time (E-T) entanglement is widely employed in long-distance quantum entanglement distribution due to its strong robustness against transmission fluctuations. In this Letter, we report what we believe to be the first silicon monolithically integrated E-T entanglement system, which integrates the photon sources, wavelength demultiplexers, and Franson interferometers on a single chip. Also, by utilizing low-loss multimode waveguides in Franson interferometers, we measured an on-chip quantum interference visibility of 99.66% (±0.47%), to our knowledge one of the highest values for integrated E-T entanglement systems reported to date. The quantum interference after 1- and 5-km fiber propagation shows visibilities of 96.72% (±0.78%) and 97.46% (±1.23%), respectively. These results demonstrate the potential of using silicon monolithic integration for advance E-T entanglement-based quantum communication networks.