With Manchester United having cut the gap to five points and Chelsea, Arsenal and Liverpool winning, any City stutter would have been cause for celebration. But Pep Guardiola's side thundered on with a club record 12th successive league victory.
Former Chelsea midfielder Oriol Romeu had cancelled out Kevin De Bruyne's opener in the 75th minute but six minutes into stoppage time Sterling, for the third match running, snatched City's winner, this time with a sublime curling shot.
Manager Antonio Conte was sent to the stands after a verbal meltdown at Stamford Bridge as his Chelsea side laboured to a 1-0 defeat of struggling Swansea City at Stamford Bridge. Antonio Rudiger's first Premier League goal sealed victory.
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"I have apologised because I did the wrong thing," Conte said of his angry reaction to a fourth official when his side were denied a corner in the first half.
"I have been in to see them after the game. It's right to speak and apologise with them."
Arsenal struck three goals in four second-half minutes as they thrashed Huddersfield Town 5-0 with Mesut Ozil scoring one and supplying assists for Olivier Giroud, who scored a brace, and Alexis Sanchez.
Alexandre Lacazette's early goal had separated the sides before the interval although the French striker went off with a suspected groin injury and will probably miss Saturday's home clash against Manchester United. Arsenal have won 12 consecutive home matches.
Liverpool moved into fifth place with Mohamed Salah taking his tally for the season to a chart-topping 12 thanks to a double in a 3-0 victory at Stoke City.
After 14 games City have 40 points, with United on 32, Chelsea 29, Arsenal 28 and Liverpool 26.
HEADY SIXTH
Burnley rose to a heady sixth, ahead of a sliding Tottenham Hotspur, as Chris Wood and Robbie Brady scored in a 2-1 victory at Bournemouth - their fourth win in five league games.
"Where we are in the league, everyone will be surprised by that," Burnley boss Sean Dyche said.
Wayne Rooney scored a hat-trick, the third with a wonder goal from inside his own half, as Everton thrashed West Ham United 4-0 on a horrible return to Goodison Park for David Moyes whose West Ham side remained in the bottom three.
Sam Allardyce, who will be named shortly as Everton's new manager, watched on from the stands as caretaker boss David Unsworth signed off with a victory that lifted Everton to 13th.
Whatever happens below them, City are looking unstoppable and with 40 points from 14 games and a goal difference of 35 they have recorded the best start in Premier League history.
They were not at their fluent best and Southampton came closest to scoring before the break with Wesley Hoedt striking the woodwork for the visitors.
But Sterling's effort sparked wild celebrations at the Etihad as the former Liverpool player struck yet another late goal to take his tally for the season to 13.
He has now earned City nine extra points this season with his timely interventions and his latest effort was arguably his best, picking out the top corner with an unstoppable effort.
"Raheem is a match-winner, I am so happy for him. He is special, it was an amazing goal. It is more than important, it is beautiful," Guardiola, whose side have dropped only two points this season, said.
City are now two wins away from matching Arsenal's record 14-match winning run in the Premier League.
(Reporting by Martyn Herman, editing by Ed Osmond)
