The game meant little for United, who had already qualified and confirmed their position finishing top of Group F in advance.
As such, Ralf Rangnick made 11 changes.
Amad Diallo and Anthony Elanga were among the starters, with Luke Shaw, Aaron Wan-Bissaka and Dean Henderson among the others coming in.
Academy players Matej Kovar, Bjorn Hardley, Teden Mengi, Charlie Savage, Zidane Iqbal and Shola Shoretire were named on a reduced bench.
United started brightly at Old Trafford and Greenwood opened the scoring with nine minutes on the clock.
Overlapping left-back Shaw crossed for the 20-year-old to hit an exceptional acrobatic volley into the bottom left-hand corner.
United continued on the front foot but Young Boys were trying their luck from distance.
Dean Henderson had dealt with the visitors' hopeful attempts, but he could do nothing to deny Fabian Rieder's curling 20-yard strike into the top right-hand corner in the 42nd minute following sloppy play by the hosts.
The second half was relatively uneventful with no further goals.
Young Boys had the best of the late opportunities, but failed to snatch a winner.
There was an emotional moment in the final minutes when Charlie Savage came on to make his debut, while his father Robbie co-commentated for BT.
Robbie narrated the substitution and said: "Coming on for Manchester United - Charlie Savage for Juan Mata.
"Wow, I never believed I'd say those words here at Manchester United.
"What a proud day for my boy. All the hard work.
"What a day for me, his mother, his grandparents and most importantly for that boy there, Charlie Savage.
"So proud of him, it's just an amazing moment."