Gabriel Jesus set the champions on their way before Sterling doubled the lead after a sweeping move.
Gabriel had a goal ruled out after VAR deemed Sterling offside before Sterling made it 3-0 after another VAR check.
Sergio Aguero then scored from a twice-taken penalty before Sterling completed his hat-trick.
The twice-taken penalty was awarded after Riyad Mahrez was clipped by Issa Diop.
Lukasz Fabianski saved Aguero's initial weak attempt but the keeper did not have both feet on his line and, after another VAR check, the penalty was retaken.
Sterling said, "The most important thing was to start well and get the win. Once the second goal went in I thought we were brilliant.
"It was quite difficult in the first half. Some of our touches were sloppy but once the team got into a rhythm the passes came and more chances came.
"With the players we have here there are competitions in training every day. It's a joy to put the ball in the back of the net.
On the use of VAR, he added: "At the end of the day, as long as the decision is right, that is all that matters."
Gabriel also said: "I am so happy for the match. I feel good with this number (nine). It is the time for me to change my career."