The visitors took their time to get going, but swung the balance of the game in their favour with two goals from Paul Pogba in the final 10 minutes of the first half.
Pogba then turned provider by teeing up a third for Anthony Martial, wrapping up the points to leave United as the early leaders of Group H.
Find out how all of the action unfolded courtesy of Sports Mole's minute-by-minute updates below.
The English clubs got off to a mixed start yesterday with defeat for Spurs and victory for Liverpool - both late on - and now it is the turn of United to get their campaign underway on a plastic pitch as they bid to continue their recent revival.
We will have a look at both teams in depth a little later, but first let's check out the team news...
Well, we'll start with the visitors and the first thing to note is that there is a debut for summer signing Diogo Dalot at right-back due to the absence of Antonio Valencia, whose knee was not trusted to hold up against the rigours of a plastic pitch.
Dalot - a 19-year-old defender capable of playing on either side - has been working his way up to full fitness since his arrival at Old Trafford, but finally gets his first chance tonight.
Mourinho does appear to have settled on a first-choice centre-back duo in the absence of Jones, though, with Smalling and Lindelof once again preferred to Bailly at the heart of the United defence this evening.
Tonight is a big chance for both to prove that they deserve a more regular role in the team, with Sanchez and Mata only making the bench for tonight's game. Rashford will be suspended at the weekend - as will Matic - so it makes perfect sense for Mourinho to give them both a runout tonight.
Paul Pogba makes up that midfield trio alongside Matic, and the World Cup winner will once again have the spotlight on him going into the game. Rumours regarding his future and relationship with Mourinho mean that he is never far from the headlines, but he will be desperate to start making them for on-field matters again.
Indeed, the Belgian has six goals in his last four outings for club and country and also has a good record against Young Boys, having scored five times against them during his time with Everton - including a hat-trick on his last visit to this stadium.
Sulejmani's return is one of three changes made by the home side tonight, with Schick and goalkeeper Wolfli dropping out of the starting XI.
Von Ballmoos and Sulejmani are the others to return to the starting XI, while summer signing Mohamed Camara retains his place having scored the 120th-minute winner in the second round of the cup at the weekend.
The former Paris Saint-Germain man is joined in attack by Assale, with Young Boys deploying a classic 4-4-2 formation this evening.
The most recent of those attempts came just last season when they beat Dynamo Kiev in the third qualifying round only to lose to CSKA Moscow in the playoffs, but tonight they can finally celebrate reaching the group stages for the first time.
Young Boys have been placed alongside United, Juventus and Valencia in Group H, and the Swiss champions would do well to avoid becoming anything other than whipping boys for those teams over the next few months.
Gerardo Seoane's side are unbeaten in their last 10 matches across all competitions so far this season, winning nine of those including a perfect record of six wins from six in the Swiss top flight.
St Gallen are their nearest challengers on 10 points, while Young Boys also boast the best attack (19) and the best defence (4) in the league too - putting them well on course to retain the title for the first time in 60 years.
Young Boys also reached the final of the cup and could have sealed a first double for 60 years, only to lose to Zurich in the final. A big match against fellow title hopefuls Basel awaits at the weekend, and another win in that match would be a major statement that they are here to retain their title this season.
The Swiss side last featured in the European Cup in 1986-87, and their most recent home match in the continent's top-tier competition actually saw them beat Real Madrid, only to be knocked out in the reverse at the Bernabeu.
Young Boys have still made it further than most in the predecessor of the current tournament, though, making it all the way to the semi-finals in 1958-59, where they lost to Stade de Reims.
Victory tonight would see them string together back-to-back Champions League wins - including qualifying - for the first time since August 2010, when their second win also came at home to an English club in the form of Tottenham Hotspur.
Defeats to Brighton and Tottenham Hotspur not only saw them lose back-to-back games for the first time since September 2016, but it was also the first time that they had conceded three goals in successive games since January 2012.
The Red Devils are now bidding to win three matches in a row for the first time since April, although one area they still need to improve is defensively, having kept only one clean sheet in their five outings so far this season.
United currently find themselves eighth in the table, six points adrift of the leading duo but now level on points with Tottenham Hotspur and Arsenal, so they have not lost too much ground in the top-four race at least.
Mourinho's side had finished top of their group in their first season back in the Premier League, although the only match they failed to win in the group did come on Swiss soil when they lost 1-0 to Basel.
However, in all European competition they have won 13 and lost only three of their last 20 games - a tally helped by their run to the Europa League title in 2016-17.
Indeed, Mourinho's side have failed to even score in their last two such outings, having been on a seven-game European away unbeaten streak prior to their loss to Basel on Swiss soil last season.
It is worth remembering that the Red Devils have not been fixtures of this competition in recent years too, although their overall tally of 22 group-stage appearances in the Champions League is fewer than only Real Madrid, Barcelona and Porto (all 23).
There is sure to be a good atmosphere inside the stadium for this historic occasions, and Young Boys are in fine form, but United look like having the quality to keep them at bay tonight. I am going for an away win.
SPORTS MOLE SAYS: Young Boys 0-2 Man Utd
The Red Devils have lost on their last two visits to Switzerland in European competition, with both defeats coming at the hands of Basel - first a 2-1 loss in December 2011 and then last season's 1-0 reverse.
As previously mentioned, those two defeats have both come on Swiss soil, so United may feel as though they have something of a hoodoo to break tonight despite never having faced this particular opponent before.
Spurs won the return leg 4-0 to progress, though, and most recently Young Boys have been beaten by a 7-2 aggregate score by Everton, with Lukaku - now of United, of course - scoring five of those goals.
YOUNG BOYS STARTING XI: Von Ballmoos; Mbabu, Camara, Von Bergen, Benito; Fassnacht, Sow, Sanogo, Sulejmani; Hoarau, Assale
MAN UTD STARTING XI: De Gea; Dalot, Smalling, Lindelof, Shaw; Fred, Matic, Pogba; Rashford, Lukaku, Martial
Elsewhere, Viktoria Plzen have taken the lead against CSKA Moscow.
The ball broke to the United skipper after Fred had been tackled on the edge of the box, and he showed great footwork to shift it onto his left foot before firing it into the top corner. It just looked so easy, but that was some finish!
It is a harsh penalty, with Mbabu having very little chance to move out of the way of Shaw's cross into the box, but Pogba doesn't care - he uses his trademark run-up before placing his spot kick into the top corner. The keeper got a hand to it, but could not keep it out.
United have been nowhere near their best, but pulled clear in the closing stages of the second half and are now in complete control of the match.
The captain-for-the-night did really well to shift the ball onto his left under pressure from a defender, and his finish left the keeper no chance.
Pogba stepped up to the resulting spot kick in his trademark fashion and, despite the keeper getting a hand to the ball, he still put enough on it to fire it into the top corner.
The hosts have also had more shots than Jose Mourinho's side, but it is that crucial stat of two goals to nil which is the difference, and that is down to the quality both sides have in the final third.
Pogba has an assist to go with his two goals, striding forward from midfield to lead the counter-attack. He is spoilt for choice ahead of him and opts to find Martial, who takes his time but eventually sees his effort deflect in off the foot of Camara.
There is an element of luck about it, but Martial will not care one bit - that is his first goal since January.
Ultimately it was two goals in the final 10 minutes of the second half from Paul Pogba which turned the game in United's favour, though, and Anthony Martial added a third shortly after the hour mark to wrap up the points.
Thank you very much for joining Sports Mole for tonight's match as United ease to victory over Young Boys courtesy of a man-of-the-match display from Pogba in Bern. I will leave you with our match report, and be sure to stick around for reaction too.
From me, though, it is goodbye for now!