The England striker snatched all three points for the Red Devils by netting at the second attempt following Paul Pogba’s cross.
United had been fortunate to be level at the interval after a lacklustre first-half display.
The Cherries had come close three times inside the first 10 minutes before a corner was half-cleared to Lewis Cook. The young England midfielder played a superb through pass to Junior Stanislas, whose pinpoint right-wing cross was stabbed in by Callum Wilson.
And the visitors were lucky not to be further behind by the time Ashley Young burst down the right and played in Alexis Sanchez, who cut it back for Anthony Martial to guide home.
United improved after the break, and nearly took the lead when Young rattled the frame of the goal with a free-kick. Rashford’s goalbound follow-up was cleared off the line by Nathan Ake, before David Brooks hacked Pogba’s effort clear.
But Rashford eventually found a way through – and victory will lift United boss Jose Mourinho at the start of a week during which his side have two more away trips – to Juventus in the Champions League on Wednesday, and Manchester City next Sunday.
Man Utd labour on the road
Only a win by a four-goal margin would have allowed eighth-placed United to displace Bournemouth from their lofty position in the top six, but that did not look like happening as they were caught cold early on.
United goalkeeper David de Gea had already been called into action to block efforts by Ryan Fraser and Wilson before Cook opened up the defence and De Gea was given no chance by the in-form Wilson – who, like his manager, was nominated on Friday for a Premier League monthly awards.
While their defence was regularly threatened by the pace of Fraser and Wilson, United were disjointed in midfield, where there was a private battle between the two clubs’ record signings to dominate the centre of the park, and attack.
Bournemouth’s £25m capture Jefferson Lerma had the better of the £89m Pogba for much of the game, while Sanchez had a frustrating afternoon, often left isolated up front or beseeching referee Paul Tierney to give a decision in his favour.
But a double substitution 11 minutes after the interval improved United’s fortunes, with Rashford and fellow replacement Ander Herrera both going close.
It was the home side who were hanging on by the end, although the visitors still threatened on the break and De Gea stood firm to keep out Brooks before Rashford’s late winner.