Carroll, Suarez ease Reds past Everton
The Uruguay international was involved in the incident which saw Jack Rodwell dismissed in the first half, and then won a penalty - which Dirk Kuyt missed - to add to his growing reputation at Anfield.
Suarez even overshadowed Andy Carroll's first Premier League goal of the season. The latter's strike was the breakthrough and set his side on the way to a victory which Everton had worked so hard for so long to deny their visitors.
Rodwell given red card
Never dull, the headlines for their latest meeting were being written as early as the 23rd minute. Rodwell slid in to challenge Suarez and although seemingly winning the ball with a tackle which did not appear to be dangerous, referee Martin Atkinson immediately brandished a red card.
Prior to that Everton had produced the best chances, with Tim Cahill - passed fit after struggling with a shin injury all week - forcing Pepe Reina into a fingertip save after rising highest at the far post, and Sylvain Distin blazing a shot just over.
But momentum of play changed in an instant after the dismissal as Cahill, who had been posing a threat to the Liverpool defence in an advanced role behind Louis Saha, was dropped back into a left-sided midfield role.
Everton's high-tempo pressing play was abandoned for a deeper-lying, more measured approach and that played into the visitors' hands as they were able to control possession of the ball.
However, they struggled to make a breakthrough until Phil Jagielka brought down Suarez on the corner of the penalty area just before half-time and Atkinson pointed to the spot. Kuyt, scorer of five derby goals, stepped up but Tim Howard saved brilliantly low to his left.
Charlie Adam did beat Howard in added time at the end of the half, but his 25-yard shot cannoned down off the crossbar. Everton continued to make light of their reduced number after the break, with Saha shooting wide after being teed up by Cahill just inside the area.
Reds make advantage pay
But Liverpool were starting to create more openings and Carroll's header looked to be creeping inside the post from Stewart Downing's corner before Saha hacked away. Another Carroll effort had Howard scrambling to his left to turn around the post while Kuyt deflected Adam's quickly-taken free-kick wide.
Saha rifled a 25-yard shot just wide of Reina's right-hand post before, in the 67th minute, Dalglish sent on Steven Gerrard, making only his third substitute appearance after six months out after a groin operation, and Craig Bellamy to win him the game.
But it was not the talismanic Reds captain who made the difference but his fellow substitute. Just four minutes after coming on, the Wales international's run down the left released Jose Enrique and Kuyt cleverly ducked under the Spaniard's cross to allow Carroll to power home from eight yards.
The match was wrapped up in the 82nd minute when Suarez grabbed the second. Collecting Kuyt's knockdown, he was lucky to have Distin's clearance bounce into his chest but he made the most of it to place a left-footed shot past Howard from six yards.