Chelsea looked set to miss the chance to extend their advantage over the chasing pack, but Cahill atoned for giving away a penalty by pouncing after a tactical switch from Antonio Conte to pile more pressure on Tottenham Hotspur, Manchester City and Liverpool - all of whom play later this weekend.
With the teams level after 70 minutes, Conte - who was without star forward Eden Hazard due to a reported muscle injury - changed his much-vaunted 3-4-3 formation to a 4-2-3-1 and Chelsea responded by upping the pressure to force a late winner.
The Blues led after 13 minutes when Willian caught Lee Grant out with a near-post free-kick for his first top-flight goal since scoring a double in the reverse of this fixture in December.
After having a goal ruled out for Saido Berahino's shove on Cesar Azpilicueta, Stoke were awarded a soft penalty when Cahill pushed Johnathan Walters to ground - the forward slamming the spot-kick beyond Thibaut Courtois.
Marcos Alonso rattled the crossbar before Conte made his tactical move and it played out perfectly for the Italian thanks to Cahill's late intervention, while Stoke's misery was compounded when Phil Bardsley was sent off for a second bookable offence in second-half stoppage-time.
Chelsea led early on after Willian's free-kick beat Grant, who dived down to his near post and got a hand to the ball but failed to stop it finding the back of the net.
The visitors' dominance continued unchecked and they should have moved two up in the 30th minute, Victor Moses scooping the ball into the box for Alonso, who drilled his shot straight at Grant.
Stoke thought they had equalised when Bruno Martins Indi headed home, but the goal was ruled out as Berahino - standing in an offside position - pushed Azpilicueta away from the ball.
The hosts did not have to wait long for another chance, though. Cahill sent Walters tumbling and the referee pointed to the spot, the Republic of Ireland international drilling home the penalty himself.
Diego Costa was his usual fiery self and was involved in heated clashes with Ryan Shawcross and Bardsley before half-time, the latter earning a yellow card.
Tempers seemingly cooled during the break and Chelsea would have regained the lead in the 66th minute had the bar not come to Stoke's rescue by keeping Alonso's brilliantly-struck 20-yard free-kick out.
Conte switched to a four-man defence by replacing Moses with Cesc Fabregas for the final 20 minutes and it was a move that paid dividends.
Chelsea upped the pressure and finally broke through when Erik Pieters followed up a lapse in concentration that gifted the visitors a corner by failing to clear David Luiz's header and Cahill thumped the loose ball home.
Stoke left gaps at the back as they pushed for an equaliser and Costa hit post before Ruben Loftus-Cheek was denied his first goal of the season in stoppage-time.
Frustrations then boiled over for Bardsley, who was given a second booking for a clumsy foul on Fabregas, as Chelsea ground out a huge win in the race for the title.
Key Opta stats:
- Chelsea have lost only one of their last 22 Premier League games, winning 19 (D2).
- Gary Cahill is the first Chelsea player to score a goal and concede a penalty in a single Premier League game since Salomon Kalou against West Ham United in April 2009.
- Eight of Willian's last 19 goals for Chelsea in all comps have been from direct free-kicks.
- Cesar Azpilicueta made his 150th Premier League appearance for Chelsea, becoming the first Spaniard to reach this milestone for the Blues.
