Everton had looked to be on course for their first derby success since October 2010 thanks to a Romelu Lukaku double in the second half at Goodison Park, but Sturridge headed home from a superb Steven Gerrard delivery in the 89th minute.
Philippe Coutinho was on the scoresheet after just four minutes in what was a lively opening 10 minutes, finishing from inside the six-yard box to hand his side the advantage.
Kevin Mirallas, a scorer for Belgium in midweek, was the man to bring Everton level. His first Premier League goal of the season restored parity just three minutes later.
Luis Suarez scored the game's third goal after 18 minutes, finding a gap in the Everton wall with an expertly-taken, free-kick that bent inside Tim Howard's right-hand post.
As well as sending Liverpool in ahead at the break, Suarez's goal also moved him top of the Premier League goalscoring charts with nine for the season.
Lukaku scored Everton's second equaliser after 71 before grabbing his second nine minute from time, only for Sturridge to salvage a point late on.
The home side made one change from their 0-0 draw with Crystal Palace, as Ross Barkley made his derby debut ahead of Leon Osman.
Brendan Rodgers, taking charge of his 50th Premier League game for Liverpool, handed John Flanagan his first derby start in place of Aly Cissokho at left-back, while Daniel Sturridge missed out with a dead leg.
Midfielder Joe Allen made his first Premier League start of the season after recovering from injury, with Suarez starting up front despite playing a full 90 minutes for Uruguay against Jordan in Montevideo on Wednesday.
After a frenetic opening, Liverpool made the ideal start when they went ahead through Coutinho after just 5 minutes.
The unmarked Brazilian poked home from close range after getting on the end of a Luis Suarez header from a corner - but the visitors were not in front for long.
Mirallas brought Everton level just two minutes later when Barkley's knockdown from Baines' set-piece found the Belgian, who turned the ball past compatriot Simon Mignolet.
And the goals continued to flow, as Suarez restored Liverpool's advantage after 19 minutes.
The Uruguayan, who helped his country qualify for the FIFA World Cup on Wednesday, executed a perfect free-kick from 25 yards to score his ninth goal in just seven Premier League appearances this term.
Barkley almost levelled soon after, with his curled effort forcing Mignolet into a smart save, before Mirallas threatened for Everton after beating Flanagan easily on the right.
Mirallas then caused the main controversy of the half, when his reckless challenge on Suarez was only punished with a yellow card 10 minutes before the interval.
Gerard Deulofeu, a second-half substitute for the injured Leighton Baines, had the first chance of the second half, shooting straight at Mignolet when one-on-one before Allen put wide from close range after Howard had gone to ground.
Mignolet frustrated Lukaku with two impressive saves before eventually the Everton striker drew the hosts level, sidefooting home from inside the area in the 72nd minute after a period of sustained pressure from the hosts.
Lukaku looked to have had the final say as he completed his double eight minutes from time, heading home powerfully from a corner, before Sturridge equalised late on.
Both sides then had chances to snatch an injury-time winners, with Suarez and Deulofeu threatening, but the spoils were shared after a pulsating game.
