Everton's miserable record at Anfield continued as Daniel Sturridge scored twice -- and missed a penalty for his hat trick -- in a 4-0 rout for a Liverpool side that proved far too strong for their neighbours.
Liverpool took the lead on 21 minutes as skipper Gerrard leaped highest to head home. Romelu Lukaku was then stretchered off with an ankle injury, and Everton's night continued to deteriorate rapidly as Philippe Coutinho set up Daniel Sturridge to make it 2-0, and Sturridge added a third with an instinctive lob to all but wrap up the game inside 35 minutes.
It got no better for Everton at the start of the second half, as they were caught on the counter-attack by a blistering break from Luis Suarez who finished superbly. It could have been 5-0 soon after as Raheem Sterling was ruled to have been fouled by Tim Howard. However, Sturridge's penalty effort for his hat-trick did not even trouble the goalkeeper.
The result means Everton are still without a Premier League win at Anfield since 1999.
Arsenal opened the door for Manchester City and Chelsea to overtake them at the top of the Premier League table as they were held to a 2-2 draw at Southampton.
The result means that victories for City against Tottenham or Chelsea against West Ham - both on Wednesday night - could move them top of the league and drop Arsene Wenger's men as far down as third place.
Southampton dominated possession in the early stages and finally made the breakthrough on 21 minutes when Luke Shaw whipped in a fantastic cross to the backpost which set up Jose Fonte with the perfect chance to score with a well-placed header.
Wenger was visibly furious as his team walked off at half-time, and they immediately sparked into life in the second half with an equaliser by Giroud on 48 minutes from Bacary Sagna's cross. Four minutes later, Arsenal were ahead as Cazorla smacked a left-foot shot into the net from the edge of the box but the lead lasted barely two minutes as Adam Lallana latched onto a loose ball in the box to score from 10 yards.
Mesut Ozil almost put Arsenal back in front midway through the second half as his mazy 40-yard run ended with a shot that crashed off the bar. Arsenal's night was further dampened by a red card for Mathieu Flamini as he lunged in two-footed on Lallana with 10 minutes to go.
Manchester United's record signing Juan Mata put in an inspired performance on his debut as his new side eased past Cardiff City 2-0 at Old Trafford.
All eyes were on Mata and the returning Robin van Persie - and the pair were involved in United's opening goal in the sixth minute.
Mata spread the ball wide to Patrice Evra with a delicious cross-field ball. Evra found Ashley Young, whose cross was headed onto the bar by Antonio Valencia.
It fell kindly for Van Persie, and the Dutchman's initial effort was saved by David Marshall. But the goalkeeper's parry only found Van Persie again, who headed into an empty net.
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Cardiff, managed by former United legend Ole Gunnar Solskjaer, came out fighting in the second half, but the hosts doubled their lead against the run of play through a fizzing 25-yard strike from Young.
The result keeps Manchester United in seventh, six points behind Liverpool in the final Champion League spot.
Crystal Palace moved five points clear of the relegation zone thanks to a 1-0 win over Hull City at Selhurst Park.
There was an element of fortune in Palace's goal as Jason Puncheon lashed home a rebound after his initial header from Yannick Bolasie's cross rebounded back to him off strike partner Marouane Chamakh.
Hull midfielder Tom Huddlestone ended the match playing in goal after Allan McGregor was sent off for kicking out at Stuart O'Keefe.
Swansea City also put daylight between themselves and the bottom three with a 2-0 victory over fellow strugglers Fulham at the Liberty Stadium.
After a dire first half, Swansea started the better after the break and took a deserved lead when a 25-yard strike from Jonjo Shelvey took a wicked deflection off Steve Sidwell leaving goalkeeper Maarten Stekelenburg wrong-footed.
The points were safe 15 minutes from time when a whipped free-kick from Jonathan De Guzman was glanced in by Chico Flores.
Elsewhere, Norwich City and Newcastle United played out an entertaining 0-0 draw at Carrow Road. Loic Remy hit the woodwork either side of half-time before being shown a red card for a scuffle with Bradley Johnson - who was also sent off.