Henrikh Mkhitaryan gave United the lead on 10 minutes with a low effort from 12 yards, before Anderlecht levelled from Sofiane Hanni's close-range rebounded effort on 32 minutes.
United, and Zlatan Ibrahimovic in particular, missed a host of chances in the second half and extra-time, but Rashford got the winner in the 107th minute with a left-footed effort following a superb turn inside the box.
Jose Mourinho's side will find out who they face in the last four in the Europa League draw at midday on Friday, which you can follow live on SkySports.com.
