With 10 goals in two matches and six points, Real is firmly in control of Group B, ahead of Juventus with two points.
Bayern also has a perfect record at the top of Group D, while CSKA Moscow pulled level with City on three points with a 3-2 victory over Viktoria Plzen, with Zoran Tosic scoring one goal and setting up another.
Bayern took a 1-0 halftime lead after City goalkeeper Joe Hart let Franck Ribery's shot slip through his grasp in the seventh minute, before Thomas Mueller and Robben piled on the misery with goals in a four-minute span just before the hour.
On a sobering night for City, substitute Alvaro Negredo grabbed a 79th-minute consolation before Bayern had defender Jerome Boateng sent off for a professional foul on Yaya Toure with four minutes left. United stayed true to its stuttering start in its first season after Alex Ferguson by conceding a late equaliser in Donetsk. Both teams have four points in Group A.
In the group's other match, Bayer Leverkusen substitute Jens Hegeler scored from a free kick in injury time for a 2-1 victory over Real Sociedad.
Paris Saint-Germain also has six points at the top of Group C, with Benfica and Olympiakos both on three points.
Top results
S. Donetsk 1 Manchester United 1
Real Madrid 4 FC Copenhagen 0
Bayern Munich 3 Man. City 1
- AP