Quick-fire early goals from Raheem Sterling and Luis Suarez put the visitors in control at Carrow Road, before Gary Hooper replied for fourth-bottom Norwich in the 54th minute.
Sterling appeared to have made the points safe with a deflected shot minutes later, but a late Robert Snodgrass header obliged Liverpool to endure a nervy last 10 minutes.
It was Liverpool's 11th consecutive league victory and means that they are guaranteed to qualify for next season's Champions League, but Brendan Rodgers's side now have a bigger prize in their sights.
They need seven points from their remaining three games to win the league and can eliminate Chelsea from the title race by winning at home to Jose Mourinho's side next weekend.
"It was massive. We said before the game, every victory at this stage of the season, whether you're top, middle or bottom, is important, and I thought we showed great courage today ,” Rodgers said.
"We will go into the next three games looking to perform well. We want to continue to fight and now look to Chelsea next week, where it will be an incredible atmosphere at Anfield."
Chelsea's shock 2-1 defeat at home to Sunderland on Saturday, coupled with Manchester City's draw against the same side in mid-week, had given Liverpool a huge opportunity to stamp their authority on the title race.
Rodgers was without the injured Daniel Sturridge and the suspended Jordan Henderson, but his side swiftly set about settling their supporters' nerves by taking the lead in the fourth minute.
Sterling collected the ball 30 yards from goal, shifted it inside Bradley Johnson, and then unleashed a 25-yard drive that flew inside the left-hand post via a slight deflection off Michael Turner.


Latest News from Sports News Desk