The hosts had dominated throughout but had only the one goal to show for their efforts when Daniel Agger handled former Reds midfielder Charlie Adam's free-kick three minutes from time.
Jonathan Walters fired low to Mignolet's right but the Belgium international reached that and substitute Kenwyne Jones' follow-up to seal a 1-0 win and get his Liverpool career off to the best possible start following a B#9million move from Sunderland.
Sturridge had only trained for two weeks having been sidelined by an ankle injury sustained at the end of last season on England duty, but if he felt rusty he did not show it.
He had already had the ball in the net once, heading in Steven Gerrard's free-kick only to be denied by an offside flag, before he eventually broke the deadlock in the 37th minute.
With most of the focus over the summer being on the future of Liverpool's other striker Luis Suarez, Sturridge completed his rehabilitation in virtual anonymity.
He is not one for creating headlines but knows where to make an impact when it matters and he ultimately settled the contest, although the game should have been put well beyond Stoke with the number of chances created.
Liverpool need someone other than Suarez, last season's 30-goal striker, to rely on if they are to seriously mount a challenge for the top four and Sturridge looks like he is ready to shoulder that responsibility.
His 12th goal in 17 appearances since joining from Chelsea for B#12million in January is an impressive strike rate but he will need to maintain that over the course of a whole season.
First of all, however, he needs to do it over the first six games - the duration of the remainder of Suarez's ban for biting Chelsea's Branislav Ivanovic in April.
Despite a summer of turmoil surrounding the Uruguay international there appears to be a sense his protestations about wanting to leave for a Champions League club - which resulted in him being forced to train on his own for a week - may have diminished.
Having been restored to training with his first-team colleagues on Friday, Rodgers said the 26-year-old was "back, smiling and happy" and there was definitely evidence of a broad grin as he walked around the perimeter of the pitch pre-match to a ripple of applause.
For all the problems Suarez has caused the club there is no doubt they will need his mercurial talents.
While they dominated a Stoke side trying to play more football under new manager Mark Hughes they lacked a cutting edge for long periods.
In fact, they 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 Asmir 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 Jordan 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.
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.
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