Jose Mourinho's side were lucky not to be behind after a sluggish start against the Swiss champions, but World Cup winner Pogba handed them an undeserved lead after 35 minutes with a superb finish from 16 yards.
He followed that up with a cool penalty as United – who handed a debut to Diogo Dalot at right-back - went into the interval with a two-goal lead.
Pogba then turned provider in the 66th minute, feeding Anthony Martial for his first goal since January 20, as United got their Champions League campaign off to a resounding start.
Young Boys almost took a stunning lead after just eight minutes, but Guillaume Hoarau's header from 12 yards flashed just wide of David de Gea's post.
The United goalkeeper was called into action again to tip over Mohamed Ali Camara's fizzing long-range drive before Marcus Rashford clipped an effort against the outside of the post as United belatedly woke from their slumber.
Pogba netted his first 10 minutes before the interval, shuffling the ball into space just inside the penalty area and steering a fine finish into the roof of David von Ballmoos' net.
He grabbed his second in the 44th minute, stroking home a spot-kick after Kevin Mbabu was harshly judged to have deliberately handled Luke Shaw's cross in the area.
Pogba was proving too hot for the hosts to handle and set up United's third shortly after the hour, laying the ball into the path of Martial, whose right-foot effort from 10 yards was deflected past Von Ballmoos by Camara.
Substitute Juan Mata whipped a late effort narrowly past the post, while Romelu Lukaku saw a late effort denied by the offside flag, but it mattered little as a Pogba-inspired United got their Champions League campaign off to a flier.
What does it mean: Giants lead the way in Group H
Juventus' 2-0 win over Valencia, despite a sending off for Cristiano Ronaldo, means that the two heavyweights of the group are sitting pretty on three points apiece. United were far from flee-flowing against the spirited Swiss, but after weathering an early storm, they were ultimately able to puff their chests out and comfortably swat aside their less illustrious opponents.
Pat on the back: Pogba in imperious form
He has been making headlines off the pitch due to speculation over his long-term Old Trafford future, but United's mercurial Frenchman was in fine form in Bern. His first was a glorious finish and he gave the goalkeeper little chance with a clinical penalty.
Boot up the backside: Rashford struggles to make impact
Recalled alongside Martial and Romelu Lukaku up front, Rashford finishes the night firmly in the shadow of his goal-scoring team-mate. The England striker hit the post in the first-half, but never came close again in a night characterised by sloppy passing and little threat.
Key Opta stats:
- Following a run of five winless matches away from home in the Champions League, Manchester United have now won three of their last five such games in the competition (D1 L1).
- United have recorded their biggest away win in the Champions League without conceding a goal since November 2013, when they beat Bayer Leverkusen 5-0 under David Moyes.
- Paul Pogba has scored his first ever Champions League goal for Manchester United, and his first in the competition since March 2016 against Bayern Munich for Juventus.
- Manchester United's Paul Pogba scored as many goals in the Champions League in the opening 45 minutes versus Young Boys (two) as he did in his previous 37 matches in the competition combined.
- David De Gea has now registered 14 clean sheets for Manchester United in the Champions League, moving level with Fabian Barthez among goalkeepers for the club – only Edwin van der Sar (28) has managed more in the competition for the Red Devils.
United will be aiming for a third consecutive Premier League win when they host Wolves on Saturday and return to Champions League action in a fortnight against Valencia. Young Boys, meanwhile, take on Basel in domestic action on Sunday and then face a daunting trip to Juventus in their next Group H game.