Antonio Conte's men got off to a perfect start on Saturday afternoon when Willian scored directly from his free kick in the 13th minute.
Stoke were awarded a penalty seven minutes before the break for Cahill's foul on Jon Walters, which he converted to threaten to seal a point for Stoke - who had seen an earlier goal from Bruno Martins Indi disallowed.
But a late corner kick ended in Erik Pieters failing to clear the danger and, instead, he unwittingly teed up Cahill who fired in to send Chelsea temporarily 13 points clear of second-placed Tottenham, who play Southampton on Sunday as Stoke saw Phil Bardsley sent off in added time.
The big news for the visitors ahead of kick-off was that Chelsea were without their star winger Eden Hazard.
The Belgian, who appeared to be targeted by the Manchester United players during Chelsea's FA Cup quarter-final win over the Red Devils on Monday night, was left behind in London after suffering a leg muscle injury in training this week.
Hazard's absence meant Conte started Pedro on the right flank and Willian - who netted a brace in the 4-2 win over Stoke on New Year's Eve - on the left.
And it was the Brazilian who proved the difference for the visitors in the 13th minute when he fired in a free kick.
The set piece had been awarded for the Marko Arnautovic's foul on Marcos Alonso and, after stepping up to the free-kick spot, Willian opted against crossing for his teammates and instead went for goal.
The Brazilian drove his effort toward the near post, which Stoke goalkeeper Lee Grant should have kept out but instead let fly into the back of the net.
Chelsea seemed injected with confidence following Willian's goal, and could have doubled their lead when Diego Costa won the ball in midfield before releasing the goalscorer. Willian back-heeled the ball into the path of Victor Moses but his subsequent low effort was blocked.
Minutes later, Willian again combined with Moses to send the ball over the Stoke defence to Marcos Alonso - whose shot was superbly blocked by Grant.
Despite Chelsea's dominance in the match at the bet365 Stadium, momentum began to shift toward the hosts. The Potters won a corner after N'Golo Kante was forced to clear Arnautovic's cross into the box just after the half-hour mark.
From it, Geoff Cameron headed the ball to Bruno Martins Indi, who then headed it past Thibaut Courtois and into the net but the goal was disallowed - much to the home fans' annoyance.
Referee Anthony Taylor - who was roundly booed following the decision - decided the offside Saido Berahino had shoved Nemanja Matic in the build-up to the goal and duly disallowed it, although it wasn't long before Stoke did eventually score.
In the 36th minute, Walters went down in the box following contact - though minimal - from Cahill during a Stoke free kick.
He confidently dispatched the subsequent penalty, but more drama was soon to follow.
Ryan Shawcross fouled Diego Costa, who had spent much of the first half getting into one-on-one battles with the Stoke players which saw him booked in the process.
And the Chelsea striker was left further frustrated when Shawcross appeared to stamp on him as he tried to jump away from the Spain international's body but wasn't booked.
Phil Bardsley's foul on Costa moments later did see him go into Taylor's book, though, which brought to an end a fiery first half.
The second period brought little in the way of action, though David Luiz did attempt an audacious long-range effort from a free kick which Grant comfortably dealt with in the 61st minute.
Marcos Alonso was next to follow after hitting the crossbar with his effort from a free kick before Matic, who started the game over Cesc Fabregas, forced a save from Grant with his drive from 25 yards out.
Conte reacted by bringing Fabregas on for Moses, and the Spaniard played a part in the winner when he delivered the 87th-minute corner kick which David Luiz headed downward into the box.
Pieters unwittingly poked the ball into the path of Cahill, and the Chelsea defender smashed the ball into the back of the net to secure a late 2-1 win and keep the Blues on course for the Premier League title.
Things, however, ended on a sour note for the hosts as Bardsley was sent off for a second bookable offence for his foul on Fabregas.