Southampton's win meant the fight to avoid the last relegation spot also tightened. The gap between 13th and 18th is just six points after Cardiff, Newcastle and Brighton also earned crucial victories, while Fulham and Huddersfield look doomed after losing again. Last-placed Huddersfield lost at home to Bournemouth 2-0 and are 16 points from safety, while Fulham remain just three points ahead in 19th after losing at Leicester 3-1.
Sterling's opening goal was hotly disputed as he was initially flagged offside but referee Paul Tierney overruled the decision after a three-minute discussion with his linesman. Sterling was in a clear offside position when Sergio Aguero played him through on goal but Tierney apparently ruled the goal should stand because defender Daryl Janmaat then tried to clear the ball, which deflected off Sterling and into the net.
"I don't know exactly what it was but I was praying and hoping that they gave it to us and we were slightly fortunate," Sterling said.
There was nothing fortunate about the other two, though. The second goal was a simple tap-in from a metre out after City carved apart Watford's defence, with David Silva and Riyad Mahrez combining to tee up Sterling.
The third was more impressive, as Sterling collected a pass from Silva and went past two defenders before slotting the ball home.
Watford substitute Gerard Deulofeu pulled a goal back for the visitors just 17 seconds after going on, scoring with his first touch after Troy Deeney flicked a long ball into the box.
Southampton scored twice in five second-half minutes to extend Tottenham's winless run to four games in the league.
Harry Kane put Tottenham ahead with his 200th goal for club and country as Spurs dominated the first half but Southampton turned it around after the break through quickfire goals from Yan Valery and James Ward-Prowse.
Tottenham manager Mauricio Pochettino accused his players of being arrogant and complacent.
"We showed a lack of aggressiveness, lack of the hunger you need to kill the games and win," Pochettino said. "I am so, so, so disappointed. I am a little bit worried about this change from the first half to the second half — it's only one thing, it's mental. It's about complacency, it's about arrogance in a bad way."
Cardiff City's hopes of staying up were boosted by a 2-0 win over West Ham, with Junior Hoilett and Victor Camarasa scoring in either half to end a run of three successive defeats.
Two other relegation candidates also earned crucial wins. Newcastle rallied from two goals down to beat Everton 3-2, putting Rafa Benitez's team six points above the relegation zone. Brighton also boosted their hopes by beating Crystal Palace 2-1, putting them level on points with Roy Hodgson's team.
- AP