Drogba's goal meant Chelsea celebrated their 107th anniversary with a win on Roberto Di Matteo's league debut, but they were far from convincing ahead of Wednesday's Champions League clash against Napoli, despite playing almost three-quarters of the game with a one-man advantage.
That was after Stoke forward Ricardo Fuller stupidly got himself sent off for a horrible stamp on Branislav Ivanovic in his first league start since New Year's Day 2011.
That ultimately handed Chelsea victory to lift them back level on points with fourth-placed Arsenal ahead of the latter's game with Newcastle on Monday.
The Blues, who needed a three-goal victory to reclaim fourth place, should have been ahead inside four minutes, Stoke captain Ryan Shawcross going close to back-heading an own goal and an unmarked Branislav Ivanovic powering over the recalled Frank Lampard's corner.
It was a confident start from the home side, emphasised when Gary Cahill drove forward from the back and unleashed a 10th-minute piledriver Asmir Begovic palmed behind.
Making his first league start of the season, Salif Diao earned an early booking for upending Ramires, who was then inches from glancing in Salomon Kalou's cross.
Cahill had to rescue Chelsea with a block after a slip from John Terry - who sealed his remarkable comeback from knee surgery - let in Jonathan Walters, who was then hauled back by John Obi Mikel, prompting another caution.
Fuller was lucky to escape a card for tripping Ivanovic as Stoke continued to frustrate their opponents but the forward then well and truly pressed the self-destruct button in the 25th minute.
Irked by a studs-up Ivanovic tackle, he inexplicably stamped on the defender's groin, earning a straight red.
Raul Meireles dragged wide when well placed on the edge of the area as Chelsea tried to make the extra man count.
They almost did just past the half-hour when Terry's downward header from Lampard's corner bounced up on to the crossbar, while Ivanovic's looping header after a one-two with Lampard almost caught out Begovic soon after.
Di Matteo threw on Juan Mata for Meireles seven minutes from the break, immediately playing a pass that saw Marc Wilson booked for bringing down Ramires.
Lampard almost danced his way through several defenders before Ivanovic wasted a glorious 40th-minute chance, smashing against the bar after Huth's header gifted him the ball.
Ivanovic failed to emerge after the break, with David Luiz coming on at right-back.
Begovic got away with spilling an early Lampard shot before Stoke threw on Wilson Palacios and Matthew Upson for Diao and Wilson in an attempt to shake things up.
They remained ultra-defensive, with Chelsea lacking the guile to break them down, something that was a constant problem under Andre Villas-Boas.
Terry drilled narrowly wide from 25 yards before Stoke's final change on the hour mark saw Kenwyne Jones withdrawn for Cameron Jerome.
Chelsea finally forced Begovic into a save - and a brilliant one at that - when Drogba's 66th-minute free-kick was clawed out of the top corner.
However, that did not stop Di Matteo sending on Daniel Sturridge for Mikel in search of a goal, ignoring Fernando Torres.
The breakthrough immediately followed, Mata's pinpoint through ball finding Drogba, whose heavy first touch drew Begovic and allowed the striker to round him and slot home his 100th Premier League goal.
Begovic failed to hold another Lampard drive as Stoke finally began to venture from their own half.
Suddenly, Chelsea were on the ropes and Andy Wilkinson even tried his luck with a 45-yard lob.
The home side were still in control but another Terry mistake seven minutes from time let in Jerome, who flashed past the post under pressure from Cahill.
Sturridge twice shot wide from the edge of the box either side of Mata's superb 30-yard free-kick clattering the upright as Chelsea eventually eased home.