Raheem Sterling scored a sensational goal early on at Carrow Road before assisting another and netting a second after the break to give his side breathing room at the top
Raheem Sterling scored the first and last goals - the former an unstoppable 25-yarder within five minutes of kick-off and the latter a slice of well-earned luck when his shot took a deflection over John Ruddy at the end of a long, determined run through the entire home half.
Luis Suarez scored the second in the 11th minute, with Sterling key in the build-up, giving the Reds breathing room to survive a Gary Hooper strike just after the break, for which goalkeeper Simon Mignolet was largely to blame.
The Canaries kept Liverpool scared when Robert Snodgrass bested Jon Flanagan in the air to head home with 13 minutes left but Brendan Rodgers's men beat the nerves and held on for a crucial win, moving five points clear of Chelsea with three games to play.
The visitors' 11th Premier League win in a row - a new club record - puts them firmly in the driving seat for a first title since 1990, while Norwich remain just two points above the relegation zone.
Joe Allen and Lucas Leiva came in for Daniel Sturridge and Jordan Henderson, absent through injury and suspension, with Sterling reprising his role at the tip of a midfield diamond that caused City so many problems in their last outing.
Just as he did against Manuel Pellegrini's team, the in-form teenager gave Liverpool an ideal start by breaking the deadlock in stunning fashion after four minutes. Philippe Coutinho fed Sterling, who dipped inside and drove an unstoppable shot past the helpless Ruddy, with the aid of a slight deflection off Michael Turner, from 25 yards.
The visitors were playing with a verve and swagger that has often been evident during their fantastic run of form and Allen forced Ruddy into a fine save before Suarez, inevitably, got on the scoresheet with his 30th Premier League goal of the season. Flanagan fed Sterling down the left and he rolled a weighted ball across the face of goal for the Uruguayan, who evaded Turner before calmly side-footing past Ruddy from eight yards.
Norwich responded admirably and Hooper was only denied what looked a certain goal by Allen's last-ditch intervention in the area.
Having started the second half on the front foot, Norwich got back into the game nine minutes after the restart when Mignolet made a mess of a cross under pressure from Bradley Johnson and Hooper was on hand to finish from close range.
Sterling quickly restored the visitors' two-goal cushion as he capitalised on Johnson's wayward pass to run at the heart of the Norwich defence before eventually beating Ruddy with a left-footed shot that took a wicked deflection off the retreating Johnson.
Snodgrass's header 13 minutes from time gave a spirited Canaries side an unlikely glimmer of hope but Liverpool held on for a crucial three points as Mignolet kept out a late header from substitute Ricky van Wolfswinkel.