The Uruguayan made it six goals in two outings in Norwich with two goals in the first half and a third in the second to lead the Reds to their first triumph of the campaign.
Suarez kicked things off with a fine finish from the edge of the area in his 50th Premier League appearance for Liverpool inside two minutes before adding a second seven minutes before half-time when he pounced on Michael Turner's defensive error before producing a top-notch effort on the outside of his right foot.
Norwich's woes were magnified early in the second half when Suarez set up Nuri Sahin for an easy finish before a fourth goal came through the former Ajax man who sent a bouncing ball past John Ruddy from outside the area for his second treble in two seasons against the Canaries.
Steve Morison knocked home from outside the six-yard box after Pepe Reina spilled Russell Martin's attempt after the hour-mark before the Reds grabbed a fifth through Steven Gerrard who was supplied by Raheem Sterling.
Norwich pulled another goal back with three minutes remaining when substitute Grant Holt fired past Reina but is nowhere near enough as Brendan Rodgers secured his first win as manager of the Anfield club.
Carrow Road, Norwich - 26,831
Yellow cards:NOR - Leon Barnett 39' LIV - nil