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Raheem Sterling opened the scoring for the Reds before Nathaniel Clyne equalized for the visitors, but Sturridge netted the winner from close range in the 79th minute to hand Brendan Rodgers' side all three points.
Speaking after the game Sturridge admitted he is under pressure to provide Liverpool's goals this season, following the departure of Suarez to Barcelona, but insisted that he is relishing the challenge.
"There's pressure on myself to get the goals now. With Luis [Suarez] out the team and of course he scored 30 odds goals last season," the forward told Sky Sports. "It's important for me and the other lads to step up and Raheem showed today that he's got a cool head and he can finish.
"I'm glad that he's continuing the scoring into the season. He's improving all the time. It's important as a collective unit we all provide the goals."
The Reds handed debuts to Dejan Lovren and Javier Manquillo while Rickie Lambert also appeared as a substitute, and Sturridge says Liverpool is a far stronger squad than the one that finished last season.
"I think so [Liverpool have improved from last season]. With the players the manager's brought in, no disrespect to Luis, he's a great player and he will be missed, but we move on from that now," he said. "We've shown today we can win games, we can score goals and we defended well. With Raheem and other players producing we'll be okay this season."