The victory was harsh on the visitors, who dominated possession for long periods and had superior chances, but it lifts United into fourth place above Arsenal, who play on Monday.
Marcus Rashford’s 10th Premier League goal of the season, after a superb through ball from Luke Shaw, gave United a flattering lead.
And as Watford wasted further opportunities to earn their first league win at Old Trafford, Anthony Martial made them pay midway through the second half with a close-range finish.
Abdoulaye Doucoure scored a late goal to fray the home fans’ nerves further.
“It was a really nervy last five minutes,” said Solskjaer, who was appointed on a three-year-deal on Thursday.
“They made it difficult for us and it seemed as though we were still on holiday or the international break. Both teams feel they should have points.
“When I won trophies here, we didn’t play well in every game but [today] we lacked zip, urgency, and fitness.”
The victory ended a run of successive defeats for Solskjaer, and took United level on points with third-placed Tottenham, who travel to Liverpool on Sunday.
After being appointed on Thursday, having lost only one of his 13 Premier League games in charge since taking over from Jose Mourinho in December, Solskjaer asked his side to work harder and warned against complacency.
However, there was a tiredness to their play throughout, which, but for Watford’s wastefulness, would have been punished. Paul Pogba also lacked his usual influence.
Whether it was because of the international break or down to Solskjaer’s new status, it was difficult to say. Far from being a party atmosphere inside Old Trafford as Ole continued ‘at the wheel’, there were times during the game when the wheels almost came off.
The visitors had 20 shots to United’s eight, with eight of them on target, yet they were often tame. Doucoure could easily have given Watford the lead early on but fired wide from 10 yards.
Troy Deeney might also have had a case for a penalty when he was charged in the box by Phil Jones.
For all United’s failings, though, Rashford’s well-taken goal and Martial’s bundled effort, which Watford thought was offside, helped grind out a much-needed victory to keep United’s Champions League hopes on track.
Their goals, in addition to those previously from Romelu Lukaku and Pogba, also ensured United have four league goalscorers in double figures for the first time since the 1995-96 season.
