Brendan Rodgers' side claimed their first three points of the season with the Uruguayan striker once again running riot at Carrow Road leaving Chris Hughton still without a win
Luis Suarez opened the scoring inside two minutes before adding another just before the break to give Brendan Rodgers' side a comfortable lead at half-time.
The Uruguayan then completed his hat-trick after the break before Nuri Sahin netted his first league goal for the Reds. Steve Morison and Grant Holt then managed two consolation efforts for the Canaries either side of Steven Gerrard's deflected strike.
It was the visitors who got off to the perfect start, and were a goal up in just the second minute with their first shot on goal. The ball popped to Suarez on the edge of the area after an intervention from Michael Turner and the Liverpool striker made Norwich pay with a good first touch followed by a smart finish past John Ruddy into the bottom left corner.
It took eight minutes before Norwich created their first opening. Jackson slid on to a cross to force Pepe Reina into a fine save at his near post with the Spaniard keeping the ball out with the help of the woodwork.
Rodgers' side felt they had a penalty in the 22nd minute after Leon Barnett bundled over Suarez in the area. However referee Mike Jones waves appeals away, which looked a harsh decision.
Jackson wasted another good chance for the hosts soon after, blazing his volley from inside the box well over the bar.
Suarez then doubled Liverpool's lead on the 38th minute, just missing a one-on-one moments earlier, he beat Turner with ease and bent a fine finish with the outside of his right boot past Ruddy to go into the break with a two goal buffer.
Norwich should have pulled one back straight from the second half kick-off, when Andrew Surman's shot across goal was somehow scooped over the bar by Robert Snodgrass a yard out from goal.
They were made to pay for the wastefulness immediately as Suarez turned provider and squared a ball across the box for Sahin to simply stroke into the goal in the 48th minute.
And 10 minutes later Suarez completed his hat-trick. Sahin found the Uruguayan cutting in from the left and the striker got his third with a curled effort round Ruddy, with the help of a small deflection.
Morison did manage to pull one back for the hosts on the hour. The Wales striker latched on to a loose ball in the box after Reina made a couple of quick saves but he parried towards the striker, who fired home on the half-volley.
Yet Liverpool restored their four goal lead via their captain Gerrard eight minutes later. He stroked home Sterling's low cross with his shot taking a heavy deflection, completely wrong footing Ruddy on its way into the net.
Suarez could have even made it six in the 72nd minute with his jabbed effort from close range only trickling a yard or so past the post.
Liverpool dominated possession for the remainder of the match and were coasting to the final whistle when Holt capitalised on Skrtel's miscue to bend a fine low shot past Reina to add another for the hosts, but it was little consolation on a dismal day for the hosts.