Success at the Riverside Stadium, however, lifted them above struggling Arsenal into fifth.
As well as the possibility of finishing in the top four, they have a second chance of competing in next season's Champions League by winning the Europa League, for which they are the bookmakers' favourites.
Upset at having to play three games in seven days, United's manager had gone as far as predicting that they would "probably lose" at Middlesbrough, but when the mind games stopped and the football began his weakened team did him proud.
"It's fantastic because we managed in a week to go to the quarter-final of the Europa League, which is an important target for us, and got these three points which keep us in the race for fourth position," he told BT Sport.
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"With injured players, tired players, suspended players, the guys were magnificent.
"We controlled everything until our second goal."
Mourinho and his coaching staff will now have a rest, even if many of their players will be away on international duty.
United resume at home to West Bromwich Albion on April 1, which is the first of eight projected fixtures that month, including two Europa League ties against Anderlecht.
Middlesbrough, in contrast, are left seeking a first league win since December, with crucial relegation games coming up after the break away to fellow-strugglers Swansea City and Hull City.
Their temporary manager Steve Agnew, pleased by his team's efforts on Sunday, was given a boost before the game when chairman Steve Gibson told the club's website(www.mfc.co.uk) "I hope he’s here this time next year or the year after. He has a lot of qualities."
Agnew, who was also briefly in charge seven years ago, confirmed that he would like the job on a permanent basis.
"It's a club I have a big affection for," he said.
"I love the people in the area and the passionate crowd. It's a pleasure to be in charge of a team and see them respond like that."
(Editing by Toby Davis)
