The Uruguayan had scored seven goals in his past three outings against the Canaries - including two hat-tricks - and bettered that by putting together a treble of goals by half-time before adding a fourth in the second term, becoming the first Reds player to net three hat-tricks against the same club.
Raheem Sterling added a very late fifth for the Reds after Bradley Johnson had pulled one back for the visitors in the 83rd minute as the Merseysiders remained fourth on the table, seven points in arrears of league leaders Arsenal.
Suarez opened the scoring in the 15th minute with a stunning effort from 35 yards over Norwich keeper John Ruddy before volleying home Philippe Coutinho's corner with his left foot from close range prior to the half-hour mark.
The former Ajax star then completed his trio of goals ten minutes before the break when he flicked the ball over Leroy Fer before firing low from about 20 yards and he wasn't finished there, scoring a fourth in the 74th minute with a sublime free-kick.
Johnson managed a consolation for Norwich with seven minutes remaining when heading home Nathan Redmond's cross before Sterling completed the rout by finishing a pass from guess who .Luis Saurez.
2013/14 Premier League scores
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