Ole Gunnar Solskjaer's side are unlikely to catch City, but Greenwood's late winner puts a little pressure on their bitter rivals with six games left.
Greenwood had opened the scoring early in the second half at Old Trafford before James Tarkowski equalised moments later. Greenwood's second made him the joint-top scoring teenager in Premier League history for United, level with Wayne Rooney on 15 goals.
Edinson Cavani grabbed the third goal in stoppage-time as second placed United won six consecutive Premier League games for the first time since January 2019. Although they have probably left it too late to overhaul City, United are on course for a second place finish and could still win the first silverware of Solskjaer's reign.
United were beaten in three of their opening six league matches this season but have lost only one of their subsequent 26.