Jordan Henderson broke the deadlock in style in the 26th minute and Suarez doubled Liverpool’s lead 10 minutes later to extend his record against Norwich to seven goals in three games.
Sturridge made it 3-0 after the interval before Steven Gerrard added a fourth, while there was also a Ryan Bennett own goal as the Reds moved closer to the European places and left Norwich without a win in six games.
Sturridge was brought into the starting line-up after scoring as a substitute in last weekend’s loss to Manchester United, and he looked eager to impress as Liverpool began brightly.
Stewart Downing looped a header over the crossbar at the back post following good work by Suarez, while Mark Bunn made a comfortable save from Sturridge’s low 12-yard effort when the new Reds striker could perhaps have played in Glen Johnson.
It was Norwich who carved out the best chance of the opening 20 minutes from a simple set-piece as Robert Snodgrass floated in a free-kick and an unmarked Bennett headed tamely at Brad Jones, starting in place of the injured Pepe Reina, when he really should have scored.
Liverpool continued to boss possession but were finding it difficult to break down an organised Norwich side, until Henderson conjured up something special when he pounced on a loose ball outside the area and smashed a powerful first-time strike into the top corner.
Full of confidence, Henderson attempted an even more ambitious volley from further out which stung the palms of Bunn, but it was a more subtle piece of play which led to Liverpool’s second.
Sturridge deliberately let a Lucas Leiva pass from deep go between his legs, bamboozling the Canaries defence and allowing Suarez the freedom to race through on goal and tuck a calm finish into the bottom corner.
Grant Holt had a shot from a tight angle blocked by Andre Wisdom but there was little for the visiting fans to get excited about as Liverpool ended the first half in complete control, with Sturridge and Suarez showing glimpses of an effective partnership.
Liverpool full-backs Johnson and Wisdom had good attacking moments in the early stages of the second period, while the lively Downing also drilled a fierce effort narrowly wide from distance.
Sturridge then got his name on the score sheet for a third successive game since making the move to Anfield, tapping in from close range just before the hour mark following an excellent right-wing cross by Downing.
Captain Gerrard fired in a trademark fourth on 66 minutes with a fine strike from 25 yards which arrowed into the bottom corner, and Norwich’s misery was compounded eight minutes later when Bennett got the final touch to a shot by substitute Raheem Sterling.