It was an exciting match from the start and both sides had been denied by the woodwork by the time Sturridge broke the deadlock in the 37th minute with a powerful low strike from outside the area which fizzed into the bottom corner.
Asmir Begovic had previously made a number of good saves to keep Stoke on level terms and continued to excel in the second half as Liverpool failed to make the most of several good chances in a manner very reminiscent of the past two seasons.
There was still plenty to admire about Liverpool's fluency even in the absence of the banned Luis Suarez, as Begovic wonderfully denied Jordan Henderson and Steven Gerrard.
There were few signs of encouragement for Mark Hughes in his first game in charge of Stoke but Liverpool's wastefulness and the brilliance of Begovic meant the Anfield faithful could never relax and it looked like there would be a late twist when Daniel Agger's handball presented the visitors with a penalty a couple of minutes from time, but Mignolet dived to his right to push away Jon Walters' spot-kick and spark the celebrations.
Liverpool could have been in trouble after just eight minutes when Mignolet, on his competitive Anfield debut, flapped at a cross with only the crossbar sparing his blushes by keeping out Robert Huth's volley.
But the tide quickly turned and Sturridge's disallowed goal was followed by Kolo Toure, another new signing, heading Gerrard's corner against the woodwork.
Sturridge blazed the rebound over from close range and then saw a cross-shot hacked clear by Erik Pieters after Philippe Coutinho had threaded through a pass which allowed him to go around Begovic.
Coutinho's vision to pick out opponents saw him set up Iago Aspas to shoot at the goalkeeper, who then blocked Jose Enrique's shot after a swift interchange between himself and Aspas.
The 21-year-old Brazilian seems certain to be Liverpool's creator-in-chief this season and he was at it again, putting Henderson through only for Begovic to get out quickly to block.
But just when it seemed Liverpool would go in frustrated, Sturridge cushioned an Aspas pass before drilling a low shot through the legs of Huth to finally beat Begovic.
One goal has never been a comfortable margin for Liverpool in recent seasons and they needed Mignolet to save well from Walters and Lucas Leiva to clear Ryan Shawcross' effort off the line to hold the half-time lead.
Further chances came after the break with Coutinho curling a shot past the far post, Sturridge denied at the near post by Begovic, Aspas heading wide from a corner and Henderson striking the upright.
Late on the excellent Begovic had to be at full stretch to tip over Gerrard's dipping free-kick and scrambled well to keep out Glen Johnson's close-range shot.
Charlie Adam almost embarrassed Mignolet with a shot from the halfway line before the goalkeeper came to Liverpool's rescue late on with his spot-kick save.
Agger handled former Reds midfielder Adam's free-kick three minutes from time. Walters fired low to Mignolet's right but the Belgium international reached that and substitute Kenwyne Jones' follow-up to get his Liverpool career off to the best possible start following a £9million move from Sunderland.