Cristiano Ronaldo sent the Red Devils 1-0 ahead in the 13th minute of the contest, but Young Boys levelled the scores in the 66th minute through Moumi Ngamaleu before Theoson-Jordan Siebatcheu scored the winner for the home side in the 95th minute.
Solskjaer has now suffered seven defeats in 11 Champions League matches, with the club losing each of their last three in the competition against Paris Saint-Germain, RB Leipzig and Young Boys.
Man United's last victory in the European Cup was a 4-1 home success over Istanbul Basaksehir last November, while Solskjaer has also overseen wins over Leipzig and Paris Saint-Germain (on two occasions).
The defeat has already left the 20-time English champions playing catch-up in Group F, and there will be pressure on the team to pick up a result in their next European fixture at home to Villarreal on September 29.
The Red Devils currently sit top of the Premier League table, though, and will be looking to continue their strong domestic form when they travel to West Ham United on Sunday afternoon.