Doigo Jota fired Liverpool in front in the 26th minute, firing a low drive past Norwich keeper Tim Krul to stake Jurgen Klopp's side to a half-time lead.
Liverpool doubled their lead in the second half when Roberto Firmino redirected a Mohammed Salah assist past Krul in the 65th minute to make it 2-0.
Salah put the Reds up by three soon after, settling a poor clearance from Norwich after a corner and curling a lovely shot into the back of the net to seal the three points for Liverpool.
With defensive linchpin Virgil van Dijk back after nine months out with a serious knee injury, Liverpool enjoyed a relatively trouble-free evening.
"It was a tough game for many reasons and 3-0 sounds comfortable," Van Dijk said after the game. "It's about getting into normal habits, shouting at your teammates, repetition. I've made great steps in pre-season and this was another big step today."