The striker opened the scoring on 19 minutes, only for Jermain Defoe to hit back from the penalty spot just six minutes later after Ragnar Klavan's foul.
Sadio Mane restored the Reds' advantage with less than 20 minutes left, but Defoe equalised a second time, beating Simon Mignolet from 12-yards after Mane's handball.
Liverpool's frustrations were compounded by an injury to goalscorer Sturridge in the closing stages, the England international's first start since 17 October ending disappointingly.
James Milner also sustained a knock, having to be replaced by Alberto Moreno at left-back at half-time due to a calf problem at the Stadium of Light.
Reds must wait on Sturridge and Milner
Klopp later told journalists in his post-match press conference that Sturridge had to go off with "a knock on his right ankle" but hoped: "I don't think it's too bad."
On Milner, the German continued that the recently-converted left-back "felt like he got a knock on the calf" but "didn't know exactly if he got a knock" or not.
"So then we needed to be really careful," said Klopp, insisting that is why he and his staff "made a change."
Klopp: No date on Henderson return yet
Captain Jordan Henderson was absent for the trip to his former club after going off in the win over Manchester City on New Years' Eve with a heel complaint.
Klopp added that he doesn't "know exactly" how long he will be without the England midfielder, adding: "Hopefully it's not too serious, but the FA Cup [game this weekend] will be difficult."
Liverpool will hope to have Philippe Coutinho back for their next league clash, a mouthwatering trip to a Manchester United side who have won six successive Premier League games on 15 January.
The Brazilian has missed their last eight games in all competitions with an ankle ligament injury, having provided five goals and five assists in his 13 league appearances so far this term.
Before the trip to Old Trafford to face their arch-rivals, Klopp's charges take on League Two outfit Plymouth Argyle in the third round of the FA Cup at Anfield this Sunday - the manager expected to make near wholesale changes and utilise his squad's youngsters and fringe players.