Jurgen Klopp's side were on top for long spells but ceded a clear chance in each half to the Canaries – Lukas Rupp botching a one-on-one and Alexander Tettey hitting the post.
Mane duly helped the Liverpool attack shift through the gears and stepped up with the winner 12 minutes from time.
The lead at the summit is now a gigantic 25 points for Klopp's rentless Reds.
Liverpool dominated possession and pushed their hosts back for the majority of the opening period but created little of note, save for a pair of sighters from the all-action Trent Alexander-Arnold.
Indeed, Norwich enjoyed the best opening of the half, only for Rupp to miss a simple pass to Teemu Pukki that allowed a scrambling Alisson to claw to safety.
Todd Cantwell fired into the side netting from a tight angle in the 52nd minute amid some ragged moments creeping into Liverpool's play, although a Naby Keita piledriver tipped over by Tim Krul showed their threat remained.
Krul excelled himself with a brilliant double save from Mohamed Salah and Keita before the hour, after which Mane's introduction from the bench gave fresh impetus to the Reds' attack.
Grant Hanley was forced to hack out of his own goal mouth in the 71st minute before Norwich surged forward on the break and Tettey almost caught out Alisson, firing a snapshot against the outside of the right post.
Hanley then miscued a free header from Ondrej Duda's corner and Daniel Farke's side would regret not making the most of that flurry.
Mane got the run on a tumbling Christoph Zimmermann – the centre-back picking a woeful moment to stain an otherwise towering display - to collect Jordan Henderson's raking ball and lash home a forceful left-footed finish.
Roberto Firmino could have ensured a nerveless finale but ballooned Alexander-Arnold's drilled cross over from close range in the 86th minute.
What does it mean? Reds' record charge remains on course
Liverpool are set to break a host of Premier League records on their march to glory and 17 consecutive Premier League wins means they are now only one behind Manchester City's all-time mark from the 2017-18 season. Norwich are seven points from safety at the foot of the table and entitled to feel slightly aggrieved over having nothing to show for their efforts.
Sadio on the Mane
Klopp getting Mane back in time for the return of the Champions League represents a huge boost. The verve and invention he injected into Liverpool's efforts here was palpable and decisive.
Oxlade-Chamberlain fails to press his claims
Filling in for any of Liverpool's much-vaunted front three is a thankless task and it was no surprise to see Alex Oxlade-Chamberlain's number raised to make way for the returning Mane with half an hour remaining. The England international failed to make an impression on the contest, although a chunk of credit should go to Max Aarons, who excelled at right-back for the Canaries.
Liverpool travel to Atletico Madrid on Tuesday to resume their Champions League defence, with their next Premier League assignment coming at home to West Ham a week on Monday. Norwich travel to Wolves in eight days' time.