The hosts had already booked their spot in this season's Champions League knockouts as group winners before this game. Meanwhile, Young Boys were eliminated from the competition a fortnight ago.
The Swiss champions, though, had a shot at finishing third in Group F and extend their continental sojourn by dropping into the UEFA Europa League. However, for that to have happened, they would have had to become only the third club in history to beat United home and away in a Champions League campaign.
They also needed Villarreal to win at Atalanta in the other game, which got postponed due to heavy snowfall in the north of Italy.
Manchester United named a heavily-rotated side comprising mostly fringe and academy players. Interim manager Ralf Ragnick excluded the entirety of his starting lineup from last weekend's Premier League victory over Crystal Palace.
A brisk start to the game saw The Red Devils break the deadlock in only the ninth minute. A well-worked team goal saw Jesse Lingard, Amad Diallo and Luke Shaw get involved. Mason Greenwood converted the left-back's cross with an acrobatic finish to put United ahead.
Manchester United created further good chances for Juan Mata and Amad Diallo, but the chances went begging.
Young Boys punished the hosts' profligacy when they drew level through Fabian Rieder. The 19-year-old scored a stunning equaliser from distance that left Dean Henderson with no chance.
United started the second half on the front foot, but soon took their foot off the pedal as Young Boys started to dominate. Former Burnley goalkeeper Tom Heaton was handed his United debut at the hour mark. At the age of 35 years, 16 years after joining the club's academy, he became the oldest Englishman in two decades to make his Champions League debut.
A couple of misplaced passes by the hosts saw Young Boys fashion a few chances to win the game in the dying moments. But they lacked the requisite composure in the final third.
The draw means the Swiss side have been eliminated from European competition this season. United will now turn their attention to this weekend's game against Norwich City, where their regulars are expected to be back.
On that note, here is a rundown of how Manchester United players fared in the game:
Dean Henderson - 6.5/10
The Manchester United goalkeeper stepped up to be counted when he was needed. But there was nothing he could have done about Rieder's equaliser.
Luke Shaw - 6.5/10
Shaw provided the assist for United's opener with an excellent cross. He was caught out of position on a few occasions when Young Boys ventured forward, though.
Eric Baily - 7/10
The 27-year-old played his role well, making many last-ditch tackles and interceptions.
Nemanja Matic - 5.5/10
The Serbian midfielder was deployed in an unfamiliar position at the heart of defence. He was the weak link in the home side's backline on the night.
Aaron Wan-Bissaka - 7/10
The 24-year-old was his usual imperious self in defence, but his attacking output was a bit more limited. Fans will be sweating on his fitness update after he suffered an injury towards the end of the game.
Amad Diallo - 6.5/10
Diallo played a role in Manchester United's opening goal. He, however, fluffed his line in front of goal before making way for Shola Shoretire in the 67th minute.
Juan Mata - 6/10
The 2010 FIFA World Cup winner had a mixed display at the heart of United's midfield, misplacing many his attempted passes.
Donny van de Beek - 5.5/10
Van de Beek was given a rare opportunity to impress for 90 minutes, but he failed to make the most of it. It was his misplaced pass on the edge of the box that led to United conceding on the night.
Jesse Lingard - 6/10
The England international had a mixed performance against Young Boys. He was one of the brightest sparks for United in the first half. But his influence waned after the restart, with his mishit passes putting United under pressure.
Anthony Elanga - 5.5/10
The 19-year-old made his UEFA Champions League bow, but his inexperience on the big stage was apparent. He, however, showed enough promise to suggest he could have a run with the first team soon.
Mason Greenwood - 8/10
The England international opened the scoring with an exquisite acrobatic finish. His directness also caused problems galore for the Young Boys defence.
Ratings of Manchester United substitutes against Young Boys
Teden Mengi - 6/10
Mengi came on with 30 minutes to go, and made a crucial interception as Young Boys sought a winner late on at Old Trafford.
Shola Shoretire - 5/10
The 17-year-old made his UEFA Champions League bow in the 67th minute, but was booked for a poor tackle only 12 minutes later.
Tom Heaton - 6/10
Heaton replaced Dean Henderson with 22 minutes to go, but was not called into action between the sticks.
Zidane Iqbal - N/A
The highly-rated Iqbal came on in the last minute of regular time, but did not do enough to warrant a rating.
Charlie Savage - N/A
The teenage son of the iconic Robbie Savage made his Manchester United bow in the 89th minute, but barely had a kick of the ball on the night.