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, when 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 are two points behind Chelsea in fifth place.
AFC Bournemouth remain on 27 points in 19th.
Man Utd: 9 Jul v Aston Villa (A)
AFC Bournemouth: 9 Jul v Spurs (H)