City had peppered PSG’s goal in the first half with Ilkay Gundogan hitting the post while Achraf Hakimi headed a goalbound Riyad Mahrez strike off the line.
Mbappe drilled in at the back post after being found by Lionel Messi’s deflected pass to put the visitors ahead before City rallied, with Sterling poking home in the 63rd minute before Gabriel Jesus slotted the winner from Bernardo Silva’s classy 76th-minute layoff.
The result confirms City’s progress and top spot with 12 points from five games, while PSG will also qualify in second after Club Brugge were thrashed 5-0 by RB Leipzig.
City boss Pep Guardiola said: “PSG, what a team, what players, what could I say? We played really well. In football the results are random. Nobody knows what will happen.”
He added: “We are into last 16 and will try our best in February to go to quarter-finals.”
AC Milan moved into contention to qualify despite their run of three defeats to start the group phase, with a 1-0 win at Atletico Madrid.
In Group D, Real Madrid clinched their progress with a 3-0 win at Sheriff Tiraspol, while Inter Milan are also through after beating Shakhtar Donetsk 2-0.
In Group C, Sporting Lisbon beat Borussia Dortmund 3-1 to clinch a last 16 spot with a Pote brace, while Ajax preserved their 100 per cent record with a 2-1 win at Besiktas after a Sebastian Haller double.