Mane returned from injury by coming off the bench in the second half and brought down Jordan Henderson’s long ball before netting at the near post in the 78th minute.
Liverpool remains unbeaten in its quest to end a 30-year title drought and has now won 25 out of 26 games in the league.
With 12 games remaining, Liverpool is now 15 points from winning the title thanks to the commanding lead over Manchester City. The second-place defending champions play West Ham on Wednesday.
Norwich is facing relegation, sitting seven points from safety in last place.