Liverpool's £58m strikeforce of Andy Carroll and Luis Suarez scored the goals that settled the Merseyside derby at Goodison Park - but Everton were left complaining bitterly about Jack Rodwell's controversial early red card.
Rodwell was sent off by referee Martin Atkinson - who has shown 15 red cards since the start of last season - after 23 minutes for what appeared to be a legitimate challenge on Suarez.
Everton had started well but were then forced to mount a rearguard action with 10 men following Rodwell's dismissal.
The hosts survived when goalkeeper Tim Howard saved Dirk Kuyt's penalty late in the first half after Phil Jagielka felled Suarez.
But with Goodison Park openly directing its full hostility towards Atkinson, Liverpool went ahead when Carroll turned in his first Premier League goal of the season after 70 minutes.
And Suarez took advantage of a misunderstanding between Leighton Baines and Sylvain Distin to add a second with eight minutes left.
Everton boss David Moyes will be frustrated at the turn of events that halted his side's early momentum - but counterpart Kenny Dalglish will happily take victory on his return to Goodison Park as Liverpool manager, the place where his first spell in charge ended more than 20 years ago.
Tim Cahill was fit for Everton after suffering a shin injury at Manchester City last week - and his value was illustrated as he was swiftly into his stride and unsettling the Liverpool defence.
It was Suarez, however, who had the first chance. Jagielka's sliced clearance found Kuyt, but Suarez could only head his cross tamely into the arms of Howard.
As a frenetic opening continued, Cahill - whose robust approach had already prompted Jamie Carragher to register his displeasure to Atkinson - rose above the Liverpool defence to force keeper Pepe Reina into an athletic tip over the top.
Everton defender Distin then demonstrated neat footwork to evade a succession of challenges inside the area before sending a rising, angled effort narrowly off target.
The game was engulfed in controversy when Rodwell appeared to win the ball cleanly in a midfield challenge with Suarez. The tackle drew an angry response from Liverpool's players- Lucas in particular as Suarez writhed in agony - but it was an obvious injustice when the England Under-21 midfielder was shown the red card.
As half-time approached with Everton desperate to get into the dressing room and regroup, Liverpool were awarded a penalty for Jagielka's rash challenge on Suarez. This time there was no disputing Atkinson's decision, but Howard rescued Everton by diving low to his left to keep out Kuyt's spot-kick.
At the end of a frantic first 45 minutes, there was still time for Charlie Adam to strike the bar from 20 yards before referee Atkinson departed to a deafening chorus of jeering from Everton's fans.
Little had been seen of £35m man Carroll until just after the restart when he twice rose powerfully to meet corners, forcing Louis Saha to clear acrobatically off the line and Howard to save low at his post.
Saha had been in lively mood after he was restored to the side and gave Reina an anxious moment with a low effort from distance that flashed just wide.
It was the signal for Dalglish to contemplate changes and they came after 66 minutes when Adam and the subdued Stewart Downing were replaced by Steven Gerrard and Craig Bellamy.
Liverpool had hardly been placing Everton under relentless pressure, but they finally forced their way through with 20 minutes left. Bellamy played in Jose Enrique, and when Kuyt ducked under his cross Carroll forced home from eight yards.
Everton were, perhaps understandably, deflated at the setback and it was no surprise when Suarez added Liverpool's second after 82 minutes. Distin and Baines were involved in a mix-up as the Uruguayan advanced into the area, and he was not about to pass up the gift to shoot low past Howard.
The striker's celebrations in front of the Gwladys Street end were interrupted by a number of bottles being thrown on to the pitch but Liverpool remained in command to collect three points in comfort, with Kuyt striking the woodwork in the closing seconds.