Matija Nastasic headed the ball back over Joe Hart, who was committed on the edge of the penalty area, leaving the goalkeeper stranded as Fernando Torres pounced to clinch a 2-1 victory for Chelsea on Sunday.
In a dynamic display, Torres set up Andre Schuerrle's opener in the first half, although that was cancelled out by Sergio Aguero soon after the break.
Having previously been held to draws by Manchester United and Tottenham, Chelsea's third straight league win will raise the team's title hopes as it sits second two points behind Arsenal.
Tottenham is a point behind Chelsea after beating Hull 1-0 on Sunday, while Sunderland climbed off the bottom with a 2-1 victory over Newcastle in the northeast derby. The day's other game saw Swansea and West Ham draw 0-0.
But one of the most eagerly anticipated matches of the season so far was at Stamford Bridge as the league's biggest spending teams over the last decade met.
Torres had missed an open goal before charging down the right flank and squaring the ball into the penalty area for Schuerrle to sweep in the opener in the 33rd minute.
But Aguero levelled four minutes into the second half after running onto Samir Nasri's throughball and getting past Cahill before beating goalkeeper Petr Cech at his near post.
Torres continued to search for his first league goal since May, but he flicked a header straight at Hart and lifted the ball over at the back post.
Then Torres produced the winner with the game about to enter stoppage time — causing manager Jose Mourinho to launch himself into the crowd behind the bench to celebrate with his son and other supporters.
"Our team isn't physical, we need (Torres') physicality," Mourinho said. "The people playing behind him: Schuerrle is fragile, Juan (Mata) is little, Oscar is fragile, (Eden) Hazard is little.
"You see City, even Aguero, he's the little one but he's a powerful tank, so we need Fernando, he gives us a lot. After that, if he scores or doesn't score it's OK, we know we can count on him."
It was Chelsea's third straight win, while City has now dropped 11 points in nine games.
At White Hart Lane, Tottenham relied on Roberto Soldado's penalty in the 80th to clinch the win against Hull and return to the top four.
Soldado's third penalty of the season came after Ahmed Elmohamady was adjudged to have handled Jan Vertonghen's cross.
Tottenham had struggled to break down a spirited Hull side until then and manager Andre Villas-Boas felt the home fans weren't supportive enough,
"We played in a very difficult atmosphere, very tense, very negative and we looked like the away team but we kept our cool," Villas-Boas said. "We had to dig deep within ourselves because we weren't getting help from anybody, the stadium reflected this atmosphere, very tense and without a lot of support until the first goal."
Sunderland has its first win of the league season after edging the derby against Newcastle.
Fabio Borini, who is on loan from Liverpool, blasted in his first Sunderland goal for an 84th-minute winner after being set up by Jozy Altidore.
Steven Fletcher had headed Sunderland in front after five minutes and Mathieu Debuchy equalized from close range in the 57th.
But Borini's goal ensured Sunderland rose off the bottom of the standings, a point above Crystal Palace, just two matches into Gus Poyet's reign as manager.