Callum Wilson had given the hosts an 11th minute lead but Anthony Martial levelled before the break, setting the stage for Rashford's late victor. From an Alexis Sanchez cross, he netted his 5th in the last four games, continuing his purple patch.
"So Lukaku has not come and we made a decision to start with Alexis (Sanchez)". I look to the team we didn't press or do any of the week we did in the week.
The Brazilian midfielder had a day to forget - not least because he was hammered on social media for a ridiculous dive.
"My assistants were saying this is the best warm-up of the season". He got away with it on this occasion but keep that up and he will soon land himself an unwanted reputation.
Callum Wilson's smart finish gave Bournemouth a deserved lead, after Juan Mata had pushed out from a half-cleared corner and was caught well out of position as Lewis Cook's clever pass played in Junior Stanislas.
Defensively they looked a bag of nerves as Bournemouth relentlessly attacked down both wings.
Jose Mourinho will look to close down the unsafe Bournemouth attack.
In the end, he had nothing to worry about.
United gradually recovered the momentum, and nearly took the lead on the hour mark when Sanchez headed towards goal from Smalling's knockdown to force another save from Begovic.
Bournemouth had a remarkable double escape in the 65th when Nathan Ake threw his body in the way of a shot from England striker Rashford after Young had hit the woodwork.
Callum Wilson is a man on fire!
The early kickoff this weekend gave us a seesaw contest from the off. Bournemouth started with objective and vigour, with Ryan Fraser forcing an early save from David de Gea. To put into context as to exactly how good he has been for the Cherries, only Eden Hazard of Chelsea has been involved in more goals and assists than the Englishman (6 goals, 4 assists).
While the players and the fans were busy celebrating the late goal inside Stamford Bridge, things got out of control on the touchline as Jose Mourinho went charging towards the Chelsea dugout.
Mourinho's team have kept just one clean sheet in their first 11 league games this season compared to eight over the same spell last term and the United manager believes it is the cost of playing more attacking football.