The Cherries led when Junior Stanislas brilliantly nutmegged Harry Maguire before firing past David De Gea on 15 minutes.
Greenwood equalised on 29 minutes with a powerful finish.
Marcus Rashford then scored a penalty after Adam Smith’s handball, before Anthony Martial made it 3-1 with a sensational strike in off the crossbar just before half-time.
King scored a penalty to give AFC Bournemouth a lifeline shortly after the break, after substitute Eric Bailly handled the ball.
But Greenwood scored with a spectacular effort on 54 minutes, before Bruno Fernandes netted with a free-kick just before the hour mark.
Manchester United move one point above Chelsea into fourth place, before the Blues host Watford on Saturday evening.
AFC Bournemouth remain on 27 points in 19th.
Man Utd boss Ole Gunnar Solskjaer, said: “I’m very, very happy with today, especially with the energy and the will and intent to go forward. That’s a game for the fans: it’s one the Stretford End would have enjoyed!”