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Rashford had scored two in two since returning from the FIFA World Cup but was benched by Erik ten Hag for "internal disciplinary" reasons as United put in a blunt first-half performance.
The England international was introduced at the break and had a huge impact, firing home a 76th-minute winner with a powerful finish past Jose Sa, before he later had a second ruled out after a video assistant referee (VAR) check.
It could be a huge three points for United on the road as it leapfrogged Tottenham Hotspur into fourth, with Spurs due to play Aston Villa on Monday (AEDT).
The hosts received a huge let-off early on when an under-hit Nelson Semedo backpass put Alejandro Garnacho in on goal, only for Sa to bail out his right-back with a vital stop.
Diego Costa missed a big chance shortly after, doing the hard work to make space for a shot but sending a tame effort straight into the waiting hands of David de Gea.
Antony spurned an opportunity before the break when he saw a header saved after great work down the left by Garnacho and Tyrell Malacia.
Rashford, brought on at half-time, had the last laugh when his battling run ended with him smashing into the back of the net.
He had the ball in the back of the net in the 85th minute but the ball was adjudged to have come off his arm, and De Gea superbly kept out Raul Jimenez's header in the closing seconds as United held on for all three points.