The build-up to the Europa League match had been overshadowed by Manchester United's complaints about the playing surface and about concerns travelling fans could be targeted by Russian hooligans.
United required only a point to secure progress from Group A but Henrikh Mkhitaryan's 48th-minute strike and a late Zlatan Ibrahimovic goal separated the sides and took the Premier League club through in second place behind Fenerbahce, who won 1-0 at Feyenoord thanks to a Moussa Sow effort
Henrikh Mkhitaryan, whose father Hamlet was also a footballer, scored 41 goals in 140 appearances for Borussia Dortmund after joining them from Shakhtar Donetsk in 2013.
Bayern coach Jupp Heynckes has been replaced by ex-Barcelona boss Pep Guardiola, Spain under-21 captain Thiago Alcantara has been recruited from the Camp Nou and Germany midfield star Mario Goetze has joined from Dortmund. Dortmund have also rung the changes by signing attacking midfielders.