Carroll and Suarez settle Merseyside derby
Referee Martin Atkinson showed what appeared to be a harsh red card to Jack Rodwell for what looked to be a legitimate challenge on Suarez, with the defender winning the ball fairly, but clipping the Uruguayan as his momentum carried him forwards.
Rodwell fiercely protested his innocence and walked off reluctantly, but his dismissal helped tip the balance Liverpool's way. As well as their two goals they also missed a penalty and hit the crossbar in the first half.
Carroll scored his first league goal of the season, and only his third league goal since his 35 million pounds move from Newcastle United in January, after 71 minutes and he took it well, sweeping home Jose Enrique's cut-back into the bottom corner.
Suarez wrapped up the points for Kenny Dalglish's team when he took advantage of some slack Everton defending to score his fifth goal of the season in all competitions eight minutes from time.
Dalglish was back at Goodison as Liverpool's manager for the first time since he famously resigned after an epic 4-4 FA Cup draw there in 1991. He said his side had deserved to win.
"We were very professional. When you play against 10 it's easy to lose your concentration and be complacent and we didn't do either," he said.
"Everyone played very well, Andy's off the mark with a league goal, and although Dirk missed a penalty it did not affect him and that's great credit to him. It was a good result for us."
Liverpool had two good chances to establish a lead in the closing minutes of the first half.
The first came after 44 minutes when Everton defender Phil Jagielka tripped Suarez to concede a penalty, but Everton escaped going behind when Tim Howard dived low to his left to save Dirk Kuyt's spot-kick.
Three minutes later Charlie Adam lashed a swerving 20-metre shot that hit the bar.