Mane, returning to action after injury, came off the bench to score the only goal of the tense Carrow Road affair in the 78th minute.
Latching onto Jordan Henderson's ball into the box, the forward fired in at the near post to finally find a way through a spirited Norwich backline and gain yet another win for the Reds.
Team news
There was one change to the line-up that defeated Southampton in the last top-flight encounter, with Naby Keita coming in to make his 50th appearance for the Reds in place of Fabinho.
Mane and James Milner started on the bench after recovering from their muscular injuries.
First half
Liverpool signalled their intent in the game with just 13 seconds on the clock as Trent Alexander-Arnold fired a shot towards goal from distance that went just wide.
Roberto Firmino and Alex Oxlade-Chamberlain then followed with unsuccessful efforts minutes later.
The visitors continued to dominate possession but found clear-cut opportunities hard to come by during the first half.
In fact, it was Norwich who produced the best chance in the opening 45 minutes, with Alisson Becker on hand to keep the encounter goalless.
A long ball over the top sent Lukas Rupp through on goal and he was joined for company by Teemu Pukki. Rupp tried to play in his teammate inside the area, but the pass was expertly cut out by Liverpool's No.1.
Second half
The Reds continued asserting the pressure and they were only stopped from taking the lead before the hour mark by excellent saves from Norwich goalkeeper Tim Krul.
Keita's effort from distance was tipped over before Krul saved a Virgil van Dijk header from the resulting corner.
Then, in the 59th minute, Mohamed Salah did brilliantly to bring down a swirling ball under control and get a shot away.
That was parried out by Krul into the path of Keita, but the Norwich stopper was up quickly to bat away the rebound.
Mane and Fabinho were then introduced to proceedings, but the next chance would fall to the hosts.
Alex Tettey found space on the right and unleashed a low shot that struck the base of the near post.
However, that let-off for the Reds was soon followed by the game's opening goal.
Henderson aimed a diagonal ball towards a surrounded Mane inside the Norwich penalty area.
The No.10 managed to get on the ball, performing a swivel as he then struck a powerful low effort that finally beat Krul.
Late chances were to be had by both teams. Firmino blazed over from close range following a drilled Alexander-Arnold cross before Alisson saved from Pukki.
But Liverpool held on to secure their 25th win of the top-flight campaign.