Rashford’s close-range finish came after Anthony Martial’s effort had cancelled out Callum Wilson’s early opener for Bournemouth at Dean Court.
It was United’s third victory in their last four Premier League games, extending an upturn in form that was started by the home victory over Newcastle United when Jose Mourinho’s side recovered from a two-goal deficit.
The result will strengthen Mourinho’s belief his side are on the up, although the United manager could easily have been contemplating a defeat had the home side made more of a succession of early chances.
Mourinho had expressed his confidence before this game that United’s position in the Premier League table would look very different heading into the second half of the season.
“I think by the end of December, which is the end of the first half of the Premier League, I think we are not going to be in the position we are now. We are going to be in a better position,” he said.
If the Portuguese’s prediction is to come true, United must shake off the inconsistency that has so far undermined their efforts this season.
This was a start, particularly as Bournemouth had begun the season so impressively.
Mourinho will know, however, there must be much more to come if they are to revive their challenge for a top four place.
Romelu Lukaku was a surprising absentee, with Mourinho revealing before the game the forward had not travelled to the south coast after picking up an injury the day before.
“In training yesterday, after the press conference, he felt something that needs to be studied in the next couple of days,” Mourinho said.
“So Lukaku has not come and we decided to start with Alexis (Sanchez).”
Lukaku had started on the bench against Everton six days previously so his starting place was far from guaranteed – he has not scored in nine games for his club.
But Mourinho decided not to risk the Belgian with Wednesday’s Champions League trip to Juventus looming.
– Sustained pressure –
Sanchez got the nod ahead of Rashford, a starter against Everton, but the Chile international had little chance to shine during the opening stages when Bournemouth could have sealed the points.
The visitors struggled to cope with Bournemouth’s pacy front players and Eddie Howe’s side exploited the lack of organisation in the United backline, forcing the lead in the 11th minute after a period of sustained pressure.
Wilson was denied by David De Gea and Junior Stanislas’s shot deflected wide before United finally cracked after failing to deal with the resulting corner.
Luke Shaw headed clear but Mourinho’s side were guilty of failing to put pressure on the ball, allowing Stanislas to drill in a low cross that was turned home by an unmarked Wilson.
United finally clicked in the 35th minute when Sanchez showed his worth.
The former Arsenal forward won the ball in midfield, with the help of his arm, according to the Bournemouth players, and played it wide to Ashley Young before advancing into the box to collect the return and set up Martial for the 35th minute equaliser.
Mourinho’s side were fortunate to be level at half-time but after the restart, they improved the longer the game went on and should have claimed a decisive second goal after Young struck the bar with a free-kick.
The ball fell to Rashford whose close range shot was blocked by Nathan Ake.
However, Rashford made amends in the final seconds when he turned in Paul Pogba’s cross from close-range.