Javi Gracia made seven changes to his Watford XI but the visitors proved miserly opponents as they frustrated Pep Guardiola's defending champion during the opening 45 minutes.
The opening goal came within a minute of the restart and amid considerable offside controversy.
Sterling pounced with Watford still reeling, netting from close range in the 50th minute and completed his treble before the hour.
Watford substitute Gerard Deulofeu reduced the deficit midway through the second half – something Liverpool must now do when they host Burnley at Anfield on Sunday.
City was forced to be patient when confronted by a heavily manned defence, although Sergio Aguero should have done better when he powered wide from Bernardo Silva's 29th-minute delivery.
Daryl Janmaat completed a brilliant last-ditch tackle on Sterling and those two were involved in the game's most contentious and pivotal moment.
After prolonged discussions between the match officials, the goal was awarded despite Sterling being in an offside position when Aguero touched the ball into his path, because Janmaat played it against the England winger's shin for the ricochet into the net.
There were no qualms when David Silva and Mahrez sliced their way down Watford's left channel and Sterling tapped in the former Leicester City man's cross.
David Silva slid Sterling through to cut inside Janmaat and Christian Kabasele to complete his hat-trick and he received a rapturous reception when he made way for Leroy Sane shortly afterwards.
Gracia's changes proved more impactful, though, as Troy Deeney and Deulofeu were introduced, the pair combining with their first touches to score 21 second later – the goal offering a tantalising glimpse of what a more expansive Watford might have been able to accomplish.