Two goals to Raheem Sterling and Luis Suarez's 30th league strike of the season helped them to their 11th consecutive Premier League victory and ensured them a top four finish and qualification for next season's Champions League, the first time they have reached Europe's elite club competition since 2009-10.
They moved to 80 points, five ahead of Chelsea who suffered a shock 2-1 defeat by last-placed Sunderland on Saturday, while third-placed Manchester City, on 71 points and with two matches in hand, play West Bromwich Albion on Monday.
Norwich failed to roll over for their higher placed opponents. After Gary Hooper had scored for Norwich early in the second half, Robert Snodgrass's 77th minute goal ensured a nervy finish for the visitors who had appeared to be heading for a comfortable win at 3-1 up.
The loss keeps Norwich mired in the relegation battle in 17th place, two points above Cardiff City and Fulham and three ahead of Sunderland.
Liverpool F.C. have not finished top since 1990, two years before the Premier League began, and they can put themselves on the brink of ending that drought when they meet Chelsea at Anfield next Sunday.
Also on Sunday, Hull City and Arsenal meet in a FA Cup final dress-rehearsal, while David Moyes returns to Everton with new club Manchester United.