Despite remaining unbeaten since a humbling 4-0 defeat away to Jose Mourinho's former club Chelsea in October, they had been unable to rise any higher than sixth place.
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.
"With injured players, tired players, suspended players, the guys were magnificent.
"We controlled everything until our second goal."
Read the full article on eurosport.co.uk: Mourinho delighted as United progress up table at last