The 24-year-old suffered the problem in the last act of the game as he collided with the advertising boards to the left of the goal at the Stretford End.
Wan-Bissaka received length treatment after the full-time whistle had blown.
Eventually, when most supporters had left the stadium, he was stretchered off down the tunnel.
Ralf Rangnick revealed to BT post-match that Wan-Bissaka had: "Two knocks - one on the knee and one on the wrist."
In the game itself, Rangnick made 11 changes to his side as they had already qualified and confirmed their position finishing top of Group F in advance.
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.