City lost both Raheem Sterling and Joe Hart to injury in the game, which is also bad news for England boss Roy Hodgson ahead of the upcoming friendlies against Germany and Netherlands.
United took the lead in the 16th minute through a brilliant piece of skill from Rashford. After being played in by Juan Mata, the 18-year-old left Martin Demichelis on the deck with a neat turn before coolly slotting past Hart.
City then lost Sterling just before the half-hour mark after he picked up a knock in a challenge with Mata.
United thought they should have had a penalty in first-half stoppage time when Demichelis appeared to step into Rashford’s path, but referee Michael Oliver waved away the claims. Players from both sides clashed over the incident but no player was booked.
Hart was forced off shortly after half-time after he was involved in a collision with Anthony Martial, who was chasing down Demichelis’ poor backpass.
City started to increase the pressure looking for an equaliser as Sergio Aguero headed against the post before Wilfried Bony missed his kick in front of goal.
But there was no way back into the game as United moved level on points with fifth-placed West Ham, and to within just one point of City in the final Champions League berth.
Tottenham Hotspur reduced Leicester City’s lead at the top of the table back to five points with a convincing 3-0 win at home to AFC Bournemouth.
Spurs were in front after just 44 seconds through Harry Kane, sliding in to guide home a superb cross from Kyle Walker.
And the second goal came after 16 minutes, another for Kane with a left-footed finish from inside the box after he had been played in by Dele Alli. That goal took him clear at the top of the Premier League goalscoring chart with 21.
It was three seven minutes after the break. Kane’s long-range shot was parried by Artur Boruc, and Christian Eriksen was on hand to tap in the rebound from close range.
The home win for Spurs was never in doubt as they continue to chase Leicester in a bid for a first Premier League title.
Newcastle United scored an 83rd minute leveller to ensure Rafa Benitez’s first home match in charge did not end in defeat in the Tyne-Wear derby against Sunderland.
Sam Allardyce, once manager of the Magpies, looked to have masterminded a famous victory until Aleksandar Mitrovic struck late for a 1-1 draw.
Mitrovic and Jack Rodwell both had chances in the opening exchanges, Mitrovic firing over when found in space in the box before Rodwell saw his header tipped over.
Jonjo Shelvey and Wahbi Khazri also went inches wide with free kicks.
Jack Colback was then booked midway through the first-half, a tenth yellow of the season that means he will miss Newcastle’s next two matches and that could prove crucial in the relegation fight. He will be suspended for the trips to Norwich and Southampton.
Sunderland took the lead one minute before half-time. A corner was worked deep to Fabio Borini, whose thumping drive was only palmed up into the air by goalkeeper Rob Elliot. The loose ball fell for Jermain Defoe, who made no mistake with the first-time volley.
Newcastle almost levelled straight after the restart but Yann M’Vila cleared off the line from Ayoze Perez, then Mitrovic headed straight at Vito Mannone.
At the other end, Elliot produced a fine stop to keep out Patrick van Aanholt with his fingertips.
The Magpies did finally find the leveller in the 83rd minute as Mitrovic nodded home after Georginio Wijnaldum had stood up a cross to the back post.
Both teams remain in the relegation zone, with Norwich two points clear of Sunderland and Newcastle one more further back.
Southampton produced a superb fightback to beat Liverpool 3-2 at St Mary’s, where the visitors were two goals up inside the opening 22 minutes. Philippe Coutinho scored the opener on 17 minutes, arrowing a shot into the bottom corner from around 25 yards out after Southampton’s defence failed to close him down.
And five minutes later it was two, as Daniel Sturridge worked the space to find a shot after being fed by Divock Origi.
Saints wasted a brilliant chance to reduce the arrears after substitute Martin Skrtel pulled down Graziano Pelle in the box. But Sadio Mane, also off the bench, saw his spot kick saved by Simon Mignolet.
Mane did get his goal in the 64th minute, moments after a fine fingertip save from Mignolet to deny Oriol Romeu’s long-range shot. It was Mane’s first goal in 17 games, beating Mignolet from a narrow angle under pressure from Skrtel.
The home side levelled in the 84th minute when Pelle curled a left-footed effort past Mignolet from the edge of the box.
And just two minutes later the game was turned on its head. After a poor clearance by Mignolet, Mane timed his run between Skrtel and Mamadou Sakho to race through and win the match.
The win lifts Southampton up to seventh in the table, level on points with Manchester United who face City in the derby.