Goals from David Silva and Sergio Aguero either side of a James Ward-Prowse own goal got the reigning champions back on track, but only after a scare late in the first half.
As in the Leicester and Palace games, City opened the scoring before conceding an equaliser – this time to Pierre-Emile Hojbjerg, who was later sent off for a rash lunge on Fernandinho.
But Pep Guardiola’s side were now about to let a third match in a row slip away and recovered to take a comfortable victory.
And England winger Sterling was thrilled with his side’s ability to pull it together and respond when faced with another diffucult challenge.
"We've had two difficult results to take. We wanted to give ourselves a chance and that's what we did, not just in terms of our football but off the ball as well," he said.
"The last three games we started well but then it's been sloppy. Today, in terms of our mentality even before we got to the stadium, we were full of fighting spirit."
Sterling, who is set to face former club Liverpool in a blockbuster clash on Thursday, added: "We had a point to prove for ourselves.
"We've had two difficult results and we wanted to bounce back and we did it brilliantly. The way we ran on and off the ball was brilliant."
The defending champions enter their clash with Jurgen Klopp’s side seven points back at the top of the table, with the Reds still unbeaten in the league through 20 matches.
City are one of just three clubs to take points off of Liverpool this season, with the two clubs having played out a 0-0 draw when the two sides met back in October.