Raheem Sterling gave the Reds the lead at Carrow Road with a brilliant strike from outside the box as early as the fourth minute.
The youngster was the architect for the second as he set up Luis Suarez for his 30th goal of the league campaign.
Norwich made it 2-1 in the early minutes of the second half when Gary Hooper found the target, but Sterling's deflected effort just past the hour mark restored Liverpool's two-goal advantage.
Robert Snodgrass pulled a goal back on 77 minutes to make it a nervy end to the match for Liverpool, who held on for their 11th successive win.
While Liverpool are now five points ahead of second-placed Chelsea, Norwich are just a position and two points above the relegation zone.
With England forward Daniel Sturridge missing because of a hamstring injury and Jordan Henderson suspended following his red card against Manchester City, Joe Allen and Lucas Leiva came into the Liverpool midfield.
Suarez, who had scored 11 goals against the Canaries in the last four meetings, led the attack as the Reds looked to capitalise on Chelsea's surprise home defeat by Sunderland on Saturday.
It was Sterling, though, who made an instant impact, firing Liverpool in front after just four minutes.
The ball was worked into the middle from the left, where Sterling held it up before turning goalwards, side-stepping Bradley Johnson and, as Michael Turner also stood off, the England international gleefully cracked a superb 25-yard effort into the corner past John Ruddy.
Norwich, who had lost the last three matches without scoring a goal to plummet towards the Sky Bet Championship, needed a response.
The hosts briefly rallied as a backward header from Hooper was flicked over, but Liverpool went 2-0 up on 11 minutes.
This time, Sterling was the provider as he cut the ball into the Norwich area from the left and Suarez arrived on cue to steer it into the bottom right corner.
To their credit, Neil Adams' reshaped side did at least offer a bit of fight, with Hooper's close-range effort hacked away from a crowded six-yard box.
England Under-21 winger Nathan Redmond tested Simon Mignolet with a 25-yard drive, which the Reds keeper pushed away.
Norwich forced a spell of pressure at the start of the second half, and pulled a goal back on 54 minutes.
A high cross in from the right by Snodgrass was punched by Mignolet, under pressure from Glen Johnson, onto Martin Skrtel. The ball dropped to Hooper four yards out and the former Celtic man slotted home his first goal since Boxing Day.
Liverpool, though, remained dangerous on the break and it was 3-1 on 62 minutes, as a stray pass from Norwich midfielder Johnson was punished in clinical fashion.
Sterling picked the loose ball up just inside his own half before darting forward and riding two challenges into the Norwich penalty area. Johnson had tracked back, but as he went to block the shot, the ball cannoned up and over Ruddy into the net.
Snodgrass forced a smart save from Mignolet with a low effort from 20 yards, before Redmond's effort was deflected behind and a header from captain Russell Martin dropped onto the roof of the net.
Norwich threatened to make a game of it again on 77 minutes when Snodgrass got up above Jon Flanagan to head in a deep left-wing cross from Redmond.
Mignolet showed good reactions to save a downward header from substitute Ricky van Wolfswinkel.
Liverpool, though, saw out the closing stages without any more real scares and could have added a fourth in stoppage time as Ruddy saved from Suarez.