Pep Guardiola shuffled his pack ahead of the resumption of Champions League football, resting the likes of Kevin De Bruyne, Rodri and Joao Cancelo, but the champions produced an assured performance as their England stars fired them to a ninth win in their past 13 league games against the Canaries.
After Sterling struck a stunning opener, Phil Foden tapped home a second and Sterling doubled his own tally with a close-range header, before the winger tapped home a rebound after missing a late penalty.
Guardiola's men now have a 12-point lead over second-placed Liverpool, albeit having played two games more than Juergen Klopp's men, while Norwich's mini-revival was halted by their dominant visitors.
City almost took the lead five minutes in, when Bernardo Silva danced through the home backline before hitting the post, before Grant Hanley headed against the woodwork at the opposite end during a frantic start to the contest.
Angus Gunn produced a strong save to deny Sterling on 25 minutes, but the England man was not to be denied five minutes later, as he beautifully curled home his first goal of 2022 from the edge of the area.
Norwich were reduced to chasing the ball for the remainder of the first half, with Ikay Guendogan spurning a good opportunity as the champions monopolised possession and avoided conceding in the opening half for the 22nd time in their 25 league games this term.
City needed fewer than three minutes to strike after the break, as Foden tapped across the line at the second attempt after a bizarre goalmouth scramble.
The visitors further extended their lead when the fantastic Sterling nodded Ruben Dias' cross home, recording his ninth league goal of the season on 70 minutes.
City's man of the match crowned his terrific performance by tapping home for a hat-trick after Gunn spilled his late penalty, awarded for Hanley's foul on young substitute Liam Delap.
