The Belgium international got his head to a Juan Mata cross 25 minutes in to settle an evenly-matched contest - just his third goal in 15 outings.
Substitute Marcus Rashford struck the crossbar and Anthony Martial also missed a sitter, but the Cherries will argue that they deserved at least a point from the match.
Relive how the 90 minutes of action unfolded with Sports Mole's live text coverage below.
STARTING XI: De Gea; Valencia, Jones, Smalling, Shaw; Matic, McTominay; Mata, Lingard, Martial; Lukaku
SUBS: Romero, Blind, Lindelof, Young, Herrera, Ibrahimovic, Rashford
STARTING XI: Begovic; Smith, Francis, Ake, Daniels; Stanislas, Arter, Gosling, Fraser; King, Wilson
SUBS: Boruc, S.Cook, Surman, Afobe, L.Cook, Defoe, Ibe
Jose Mourinho: "When you lose, you probably have a little bit more desire to win. Nobody likes to lose once, nobody wants to lose twice. I think that's something that belongs to every team in the world. When you lose a match the next one, you have that little bit of extra to give."
Eddie Howe: "Man United's home record is very strong so I think you have take [the defeat to] Manchester City as a one-off. You have to relish [a trip to Old Trafford]. I always look back to our old League Two days and we'd have done anything to put ourselves in these positions."
MANCHESTER UNITED SUBS: Romero, Blind, Lindelof, Young, Herrera, Ibrahimovic, Rashford
BOURNEMOUTH SUBS: Boruc, S.Cook, Surman, Afobe, L.Cook, Defoe, Ibe