With title rivals Liverpool playing Burnley on Sunday, City struggled in a goalless first half.
Sterling ended Watford's resistance just 40 seconds into the second in fortuitous circumstances.
Sergio Aguero’s pass was cleared by Daryl Janmaat against Sterling and the ball looped in.
His second came four minutes later, tapping in Riyad Mahrez’s cross and he completed his hat-trick on 59 minutes after being played in by David Silva.
Watford got a consolation when substitutes Troy Deeney and Gerard Deulofeu combined with their first touches for the Spaniard to score.
City have 74 points, having played one more than Liverpool.
Watford remain eighth on 43 points.