Ashley Young was on target for the first time in 18 months as he scored twice for second-placed United who led 3-0 at the break before Watford produced a late fightback.
United's win moves them to 32 points and they are within five points of leaders City who host Southampton on Wednesday.
Leicester's Jamie Vardy and Riyad Mahrez turned the clock back two years as they both scored in a first-half display reminiscent of when the Foxes won the title in 2015-16.
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Harry Kane replied for Tottenham after the break but the visitors could not prevent a third defeat in five league games which left them in fifth place -- 13 points off top spot.
West Bromwich Albion's caretaker manager Gary Megson looked set for a morale-boosting victory as his side led Newcastle United 2-0 with goals from Hal Robson-Kanu and Sam Field.
But Newcastle rallied to avoid a fifth straight league defeat with Ciaran Clark halving the deficit and an own goal by West Brom's Jonny Evans frustrating the hosts who are expected to name a successor to sacked Tony Pulis this week.
The BBC reported after Tuesday's match that former Newcastle and Crystal Palace manager Alan Pardew, 56, was set to be appointed as the new West Brom manager as early as Wednesday.
Basement club Palace drew 0-0 at Brighton & Hove Albion to move level on nine points with second-bottom Swansea City.
LIKELY CHALLENGERS
Pep Guardiola's Manchester City look unstoppable but neighbours United are emerging as their most likely challengers with a derby looming next month.
United have been hamstrung by their poor away form, losing their last two on the road in the league, but they found a cure for travel sickness at a freezing Vicarage Road where manager Jose Mourinho said they could have scored seven or eight goals.
As it was they were hanging on towards the end after Tory Deeney's converted a 77th minute penalty and Abdoulaye Doucoure stabbed home to make it 3-2 with six minutes left before Jesse Lingard's superb solo goal sealed the win.
"The second half should have been five or six nil, it was easier to score than the first," Mourinho said.
"We missed chances. Out of nothing, they got a penalty for 3-1, then it was a different game. We should be smoking cigars but then we're in trouble.
"It's important we have three wins in a row, nine points in three matches is good."
Former Watford winger Young had put United ahead after 19 minutes with a powerful near-post effort, celebrating by cupping his ear to the home faithful who responded with boos.
He topped that with a superb free kick that gave goalkeeper Heurelho Gomes no chance six minutes later.
Anthony Martial made it 3-0 in the 32nd but home substitute Deeney's spot kick breathed new life into the hosts who began the night seeking a third straight league win of their own.
Needless to say their manager Marco Silva was not inclined to agree with fellow Portuguese Mourinho's verdict that United could have won by a much larger margin.
"The result is hard to take for us, we didn't deserve this result," he said.
(Reporting by Martyn Herman; Editing by Ken Ferris)
