The champions endured a frustrating first-half, but blew the visitors away with three Sterling goals inside the first 15 minutes of the second period.
It was no more than City deserved after dominating possession throughout.
City came close to taking the lead with just eight minutes on the clock as Riyad Mahrez spotted David Silva’s dart into the box and the Spaniard glanced the ball just a foot past the post with Ben Foster beaten.
The visitors set up to defend in numbers – an all-too familiar puzzle for the Blues to face, particularly of late with Chelsea, West Ham and Bournemouth all playing similar stifling tactics.
The Hornets were taking a physical approach to the game, quickly crowding out the Blues’ creative talents whenever they were around the box and attempting to counter-attack whenever the opportunity arose.
It wasn’t until the 29th minute that City finally got a good sight of goal as Bernardo whipped in a superb cross that Aguero headed a foot wide.
It felt like those chances would be few and far between if the pattern of play continued.
Sterling then raced onto a ball behind the Hornets defence, but was denied by a well-timed challenge from Daryl Janmaat – so the scores remained level, but the Blues were starting to turn the screw.
The home crowd were quickly tiring of Watford’s time-wasting tactics at goal-kicks and from throw-ins – something that would surely only be ended by a goal for the champions.
Mahrez fizzed a shot over the bar in first-half added time, but that was the last act of a half that had been a tough watch for the Etihad crowd – and the fourth first period in a row that the Blues had failed to score.
A minute after the re-start, however, and the Blues were ahead in controversial fashion.
Ilkay Gundogan’s lofted ball found Aguero who deflected the ball into Sterling’s path and as Janmaat attempted to slide the ball out it hit Sterling’s shin and looped over Ben Foster and into the net.
The assistant referee flagged for offside, but after a three-minute discussion with match official Paul Tierney, the goal was awarded, much to the chagrin of the Watford players who had seen all their hard work undone in bizarre circumstances - but the delight of the City players.
Would VAR have ruled otherwise? Who knows - but the referee was convinced it was a good goal and that’s all that mattered on this occasion.
Watford’s concentration broken, City then doubled the lead almost straight from the kick-off, this time David Silva was the architect as he slid a pass to Mahrez, and the Algerian’s low cross gave Sterling a tap-in from two yards.
Two goals in the first five minutes of the second-half – it’s fair to say few people would have seen that coming given all that had gone on before!
But it was about to get even better.
The Hornets were suddenly coming apart at the seams and just before the hour-mark, Sterling completed his hat-trick with the pick of the bunch.
After drifting behind Janmaat, he collected David Silva’s pass, drifted past a couple of challenges before dinking the ball over Foster to put City three up – it had taken him just 13 minutes in total and put him on 18 goals for the season.
He left the pitch shortly after to a standing ovation and was replaced by Leroy Sane.
Moments later, Watford made a double substitution themselves and it paid instant dividends as Troy Deeney headed Foster's long kick into the path of Gerard Deulofeu who prodded past Ederson with 66 minutes on the clock to hand the visitors a lifeline – the pair had been on literally seconds.
The Blues had good chances to increase the lead in the time that remained, but Bernardo was denied by a fine reflex save from Foster and then Sane shot wide from a good position - though a difficult angle.
As it was, 3-1 was more than enough as the Blues relentlessly power on.
"Almost impossible to attack when they play so deep -there were no spaces. It was so difficult. but we created enough chances.
"After first goal, it was easier - it is a debate for the referees. I don’t know really. VAR is ready but, nothing happens this year because the Premier League didn’t want VAR - ask them why it is not here.
"Most of us want it - some didn’t That’s why this situation happened. I don’t like it but it is what it is.
"As for Raheem, three goals fantastic but he can do better. He didn’t help us a lot defensively and he missed some chances.
"Most of the games we play are like this – it so difficult because they are so organised. We spoke at half time: make more movement in attacking movement. We did it. That’s why first goal can be a debate.
"We accept what law says. I don’t know what happened. We made an incredible performance."
What it means…
City extend the lead at the top of the Premier League to four points - Liverpool take on Burnley tomorrow afternoon.
Star man: Raheem Sterling
Not a difficult choice!
City are back in action on Tuesday evening in the Round of 16 Champions League second leg clash with Schalke.
City lead 3-2 from the first leg.
Then it’s off to South Wales for the FA Cup 6th round tie against Swansea next Saturday (5.20pm kick-off) before a welcome two-week international break for the Blues.