Michael Lang popped up at the back post to tap home in the 89th minute to give the hosts a hard-earned three points after ceding possession to Man United for much of the match.
Jose Mourinho's team had the better of the chances in the opening 45 minutes, but through a mix of bad luck, some decent defending in the final third by Basel and strong play by Tomas Vaclík in net, no goals ever came.
The hosts were far the stronger side coming out of the dressing room, though, with Renato Steffen forcing a save from Sergio Romero in the 56th minute before firing a lovely curled shot just wide of the far post minutes later.
Marcus Rashford, on for Jesse Lingard in the 64th minute, saw his shot from a Romelu Lukaku assist skip into the side netting right before Zlatan Ibrahimovic came on in the 73rd.
The result moves Basel into second place in the group -- level on points with CSKA Moscow but ahead on goal difference -- while United will need a result on their final matchday on Tuesday, Dec. 5, to ensure qualification.