Liverpool's bid for a first league victory of the season started perfectly when Suarez took control of the ball on the edge of the area before finishing well past a static Norwich defence.
Despite a good start from the visitors, Simeon Jackson was unlucky not to equalise a couple of minutes later as Pepe Reina reacted brilliantly to block the striker's effort.
The visitors had claims for a penalty rejected midway through the first half when Suarez went down after a challenge from Leon Barnett.
The Uruguayan forward did double his tally before the break as he stole the ball from Michael Turner before calmly slotting past John Ruddy.
Liverpool put the result beyond doubt two minutes into the second half, with Suarez turning provider by crossing for Nuri Sahin to apply a simple finish from close range.
Suarez completed his hat-trick with a curling effort that landed in the bottom corner of the net courtesy of a small deflection to round off a superb individual performance.
The hosts did manage to find the goal that their attacking play deserved when Russell Martin's shot was well-saved by Reina but Steve Morison finished powerfully to ruin Liverpool's hopes of keeping a clean sheet.
It was captain Steven Gerrard who completed the rout for Liverpool as his strike hit Barnett after good work on the wing from Raheem Stirling to leave Ruddy with no chance.
Grant Holt added a consolation late on for the Canaries but the visitors eased to their first victory in the Premier League this season.