Willian's opener had been cancelled out by a Jonathan Walters penalty in the first half, but Gary Cahill's late winner put his side within seven victories of being crowned champions of England for a sixth time.
Antonio Conte had set his side the challenge of picking up 24 points from their final 11 games of the season in order to secure the title, but it was quickly clear that the visitors would have to fight for it if they were to get the first three of those this afternoon.
Stoke did not let Chelsea have anything easily in the first half, but it took a moment of individual inspiration, coupled with a goalkeeping error, to put the league leaders in front after just 13 minutes when Willian caught Lee Grant out with a cheeky free kick for his first goal since scoring a brace against the same opponents on New Year's Eve.
The Stoke keeper was expecting a cross into the middle, but Willian instead curled a shot towards the near bottom corner from the left flank and Grant was unable to keep it out despite reaching the ball in time.
Grant did go some way to atoning for that error on the half-hour mark, though, making a crucial save to deny Marcos Alonso from close range as Chelsea looked to build a two-goal cushion.
Stoke thought that they had levelled things up shortly afterwards when Martins Indi planted a header past Thibaut Courtois but, after a brief discussion with his linesman, referee Anthony Taylor disallowed the goal due to a push by Saido Berahino in the build-up.
If the Potters felt aggrieved with that decision then they were on the right end of another soft one moments later when Cahill was adjudged to have pushed Walters over inside the area. Taylor pointed at the spot, and Walters stepped up himself to thump his finish down the middle.
The goal increased Chelsea's wait for a Premier League clean sheet to six games, but the visitors almost provided an immediate response when Pedro fired over from a rebound after seeing his initial shot blocked.
Chelsea continued to see more of the ball in the second half but Stoke managed to limit their sights of goal and it took a 40-yard free kick from David Luiz to test Grant for the first time after the interval.
It was via free kicks that Chelsea continued to threaten most, and they were inches away from regaining the lead shortly after the hour mark when Alonso sent one of his own crashing against the crossbar with Grant beaten.
The visitors threw wave after wave at the Stoke defence in their attempts to find a winning goal, but Grant denied both Nemanja Matic and Pedro before the hosts began to fight back with their best spell of the second half.
Courtois remained untested during Stoke's period of pressure, but there were chances for Mark Hughes's side as first Walters and then Ramadan Sobhi lacked the composure to make the most of two opportunities in quick succession.
Having raised hopes of an unlikely victory, it was instead Stoke who were on the wrong end of a late goal as Chelsea once again found a way to win when Cahill - who also scored in the reverse fixture between the two sides - swept a loose ball home with just three minutes remaining following a mistake by Erik Pieters.
Suddenly Stoke began to commit more bodies forward in an attempt to salvage a point from the game, but that allowed Chelsea to attack almost at will and there should have been a third late on when Costa fired a shot against the outside of the post before Ruben Loftus-Cheek was denied by Grant.
The hosts' frustration boiled over in the closing stages when Phil Bardsley earned himself a red card in the fifth minute of stoppage time, picking up a second booking for a late and heavy challenge deep inside his own half.
Chelsea now hold a 13-point lead at the top of the with only 10 games to go - an advantage no side has ever squandered in Premier League history.
Tottenham Hotspur and Manchester City will look to close the gap back down to 10 when they take on Southampton and Liverpool respectively tomorrow, but Chelsea continue to march on and know that, no matter what happens below them in the table, 21 points will be enough to see them crowned champions of England for a sixth time.
Stoke, meanwhile, remain ninth in the table and are now seven points adrift of the top eight following West Bromwich Albion's win over Arsenal in the early kickoff.