As has been the case in recent weeks, United made a slow start to the match and fell behind after only 11 minutes when Callum Wilson broke the deadlock for the high-flying hosts.
United were without Romelu Lukaku after he picked up an injury in training ahead of the game, with Alexis Sanchez returning to the starting XI ahead of Rashford as Jose Mourinho's only change from the win over Everton.
The visitors were sluggish in the opening half-hour, though, and Bournemouth could have broken the deadlock after only four minutes when Chris Smalling failed to cut out Wilson's pass to leave Ryan Fraser clean through on goal, only for David de Gea to deny the in-form winger.
Bournemouth continued to pour forward and threatened again after 10 minutes when first De Gea kept out Wilson and then Junior Stanislas saw a shot deflected wide from inside the area.
The hosts did break the deadlock just one minute later when United's defence went missing after a corner, allowing Stanislas to break down the right and deliver a low cross into the box which Wilson tucked away with aplomb.
Eddie Howe's side were not content to sit on their lead and almost doubled it after 19 minutes when Adam Smith cut inside and put a low cross into the box which bobbled narrowly wide of the far post with De Gea beaten.
Stanislas was the next to come close as Bournemouth continued to get joy on the counter-attack, but his curling effort from the edge of the box was too close to De Gea.
A lacklustre United did finally begin to improve after the half-hour mark and created their first chance in the 33rd minute when Fred met a low cross from the left, only to see his wayward effort blocked by Nathan Ake.
Martial was not so wasteful moments later when a similar chance fell his way, with the France international confidently stroking a Sanchez cut-back into the corner to score in a fourth successive Premier League game for the first time.
The equaliser was harsh on a Bournemouth side that had been much the better team up to that point, but United did end the first half on top and Juan Mata dragged one shot wide after a good knockdown from Sanchez.
Bournemouth did offer another reminder of their threat on the counter before the interval when Fraser stung the palms of De Gea, and the United keeper was also called into action early in the second half when he made a smart reaction stop to deny a clever David Brooks flick.
The visitors would go on to control the vast majority of the second half, though, and they squandered a big chance to gain the lead 10 minutes after the restart when Martial's flick released Luke Shaw, whose low effort was well saved by Asmir Begovic.
Sanchez - who was one of United's best players on the day - fashioned another chance out of little on the hour mark and the resulting corner was not properly dealt with by Bournemouth, allowing Ander Herrera to fizz a low drive wide from the edge of the box.
Mourinho would have been forgiven for thinking that it might not be his side's day when United somehow failed to score shortly afterwards, with Ashley Young's initial free kick crashing off the crossbar and bouncing back into a dangerous area.
Rashford and Paul Pogba both had golden chances with the goal gaping, but Ake first denied Rashford with a brilliant block before Brooks hacked Pogba's follow-up effort off the line.
Herrera came close again shortly afterwards as United's pressure grew, this time curling a left-footed strike inches wide of the far post from just outside the area.
Bournemouth did eventually stem the flow of chances and threatened a winner of their own nine minutes from time when the impressive Brooks stung the palms of De Gea, but it was United who went on to take all three points.
Begovic denied Jesse Lingard before the decisive goal, which came in the second minute of injury time when Rashford collected Pogba's cross and kept his composure in the area before firing it home from close range.
It was harsh on the hosts - who would have climbed into the top four with a victory - as they fell to their first home defeat of the season, while United have now won back-to-back games for the first time since mid-September.
Mourinho's side now face a difficult week ahead of them, taking on Juventus in Turin before the Manchester derby against champions City at the Etihad Stadium next weekend.
BOURNEMOUTH (4-4-1-1): Begovic; Francis, S Cook, Ake, Smith; Stanislas (Surman 91'), L Cook (Gosling 68'), Lerma, Fraser (Ibe 74'); Brooks; Wilson
MAN UTD (4-3-3): De Gea; Young, Lindelof, Smalling, Shaw; Matic, Fred (Herrera 56'), Pogba; Mata (Rashford 56'), Sanchez (Lingard 78'), Martial