Everton had looked on course for victory with two goals from Romelu Lukaku, but were forced to settle for a point.
Philippe Coutinho had put the Reds ahead early on before Kevin Mirallas levelled, then Luis Suarez made it nine league goals for the season from a free-kick only for strike partner Sturridge to join him at the top of the scoring charts late on.
There was huge controversy at Stoke where referee Kevin Friend sent off Sunderland's Wes Brown despite the former Manchester United defender having appeared to win the ball fairly in a challenge with Charlie Adam.
Black Cats boss Gus Poyet was incensed by the decision, which came shortly after Adam had put the home side ahead and Steven Nzonzi scored late on to secure a 2-0 victory and leave Sunderland bottom of the Barclays Premier League table.
Newcastle beat Norwich 2-1 despite a nervous finish at St James' Park. Loic Remy took less than two minutes to the open the scoring with a close-range header from Yohan Cabaye's corner, with Yoan Gouffran extending the lead against former Magpies manager Chris Hughton's side.
Leroy Fer's late header for the Canaries made it a tense finish but Alan Pardew's men held on.
The pressure on Fulham boss Martin Jol ramped up after a home defeat to Swansea.
Jonjo Shelvey and an Aaron Hughes own goal clinched the points for Swansea, while Fulham's only consolation was Scott Parker's first goal in all competitions since February 2011.
New Crystal Palace boss Tony Pulis was at the KC Stadium to watch his new club take a priceless 1-0 win over Hull despite being reduced to 10 men.
Palace, who appointed Pulis as new manager just before kick-off, leapfrogged Sunderland in the table after Barry Bannan's close-range goal, minutes after Yannick Bolasie was sent off for a challenge on Jake Livermore.
Two goals for Olivier Giroud saw Arsenal go four points clear at the top of the table after a 2-0 win over Southampton.
The first came courtesy of a howler by Saints keeper Artur Boruc, who tried to dribble past Giroud only to be robbed by the Gunners striker who scored from close range.
Giroud then sent Boruc the wrong way from the spot for his 10th goal in all competitions after Jose Fonte was penalised for holding Per Mertesacker's shirt.