The Portugal international looked threatening after scoring two goals on his debut United debut against Newcastle United last weekend.
The visitors held onto the 66th minute when Cameroon winger Moumi Ngamaleu's equalised with a close-range finish past David De Gea.
It seemed United would hang onto a point at least, but a 95th minute mistake from substitute Jesse Lingard ultimately cost them the game.
The English playmaker, who had replaced Ronaldo with 18 minutes left, attempted a aimless back-pass which allowed Jordan Siebatcheu to score a simple winner.
The home fans celebrated wildly as Young Boys secured what is undoubtedly one of their greatest victories.
Despite spending £120m on three major signings in the summer, it was the same old story for United in the Champions League following their group stage exit last season.
The Red Devils mustered just two shots for the entire game despite their wealth of firepower in attack.
Speaking after the game, Solskjaer told reporters: "We need 12 points to go through. You need to win your home games, win one away from home and go through.
"A missed opportunity tonight, of course, but then again last year we won our first two games against PSG and Red Bull [Leipzig] and everyone thought we were through. So we have five games to get the 10 or 12 points."