Rashford had already missed a golden opportunity on the rebound when Ashley Young's free-kick hit the crossbar, but he came up trumps for Jose Mourinho's men at the Vitality Stadium on Sunday (AEDT).
Paul Pogba's left-wing cross was chested down by the England forward, who kept his composure to fire past Asmir Begovic and earn United their first away league win in three attempts.
Anthony Martial earlier levelled for United, scoring for the fourth Premier League game in a row, as Bournemouth's fine start - Callum Wilson scoring an early opener - counted for nothing.
Other Premier League results:
Newcastle 1-0 Watford
Everton 3-1 Brighton
West Ham 4-2 Burnley
Cardiff 0-1 Leicester
Arsenal 1-1 Liverpool