AFCON stars Sadio Mane and Mohamed Salah also scored in what ended up being a clear win, but that far from tells the full story.
Jurgen Klopp made seven changes from the midweek Champions League win away at Inter Milan, and his side looked disjointed throughout the first hour of the match.
The first half surprisingly ended goalless despite a number of chances, particularly for Teemu Pukki, who struggled to stay onside.
However, the Canaries had some luck two minutes into the start of the second period, with a speculative shot from Milot Rashica deflecting off of Joel Matip and over Alisson Becker in the Liverpool goal.
Klopp, perhaps realising his error in making so many changes, brought on Divock Origi and Thiago Alcantara minutes later, and the difference was instantly noticeable.
Thiago upped the tempo, and in the 64th minute Liverpool's title push was back on when Sadio Mane acrobatically bicycle kicked in a Kostas Tsimikas cross to level the score.
And two minutes later an afternoon of frustration came to an end when goalkeeper Alisson fired a drop kick up to Salah, who toyed with Angus Gunn and the Norwich defence before tucking the ball home.
The Egyptian is the second fastest player to reach 150 goals for the club, doing so in 233 games, only bettered by Roger Hunt's record of 226.
The gloss was added with nine minutes to play when £37million January signing from Porto, Luis Diaz, showed his brilliant off the ball running with a dart into the Norwich box.
Reds captain Jordan Henderson spotted the move, finding the Colombian who took the ball in in his stride before capping off a superb start to his Liverpool career with a confident dink over Gunn.
The result helps Liverpool keep Premier League leaders Man City honest, momentarily closing the gap from the top of the table to six points before Pep Guardiola's side take on Tottenham.
As for Norwich, a bad afternoon will be made even worse when they take a look at results elsewhere, with relegation rivals Burnley and Watford both winning to send Dean Smith's side back to the foot of the table.