The England international, whose recent form has come under scrutiny after a series of mistakes, was at fault as he raced out of his penalty area in an attempt to reach Willian's 90th-minute pass forward.
Centre-back Matija Nastasic, under pressure from Torres, headed the ball back but it went past Hart and the Chelsea striker raced on to fire home his first Premier League goal of the season.
The goal capped one of the Spain striker's better games for the Stamford Bridge side as, after missing one gilt-edged chance, his run and cross allowed Andre Schurrle to open the scoring in the 33rd minute and before he himself crashed a shot against the crossbar.
City improved in the second half and a brilliant left-footed strike from Sergio Aguero, beating goalkeeper Petr Cech as his near post, pulled them level only for Hart's aberration to undo all their good work.
The result lifted Chelsea back above Liverpool into second on goal difference, two points behind leaders Arsenal.
A disputed late Roberto Soldado penalty ensured Tottenham stayed in touch with the top four as they beat Hull 1-0 at White Hart Lane.
Tigers midfielder Ahmed Elmohamady was adjudged to have handled Jan Vertonghen's cross and the Spain international stepped up to convert from the spot.
On-loan Liverpool striker Fabio Borini gave Sunderland their first win of the season in the Tyne-Wear derby with a quality strike six minutes from time at the Stadium of Light.
Steven Fletcher's fifth-minute header considerably lifted the mood of the Black Cats, who began the day bottom of the table, but they were deflated when Mathieu Debuchy slotted home just before the hour.
Borini was sent on as a 69th-minute substitute and cut in from the left to smash Jozy Altidore's lay-off past Tim Krul.
Swansea moved up to ninth after a goalless draw at home to West Ham.