The England international scored right on half-time as the Red Devils made it six successive wins in all competitions since Ole Gunnar Solskjaer replaced Jose Mourinho in the Old Trafford hotseat a month ago.
Spurs dominated the second half at Wembley but spurned numerous chances while David De Gea was in majestic form to ensure United came away with the three points which moves them level on 41 points with fifth-placed Arsenal and six behind Chelsea in fourth.
The result also damages Spurs’ title aspirations as a second successive home defeat leaves them third and nine points behind leaders Liverpool.
Hugo Lloris replaced Paulo Gazzaniga in goal, while Jan Vertonghen started in place of Davinson Sanchez at centre-back and Ben Davies came in at left back in place of Danny Rose as Mauricio Pochettino made three changes from the Tottenham side that beat Chelsea 1-0 in the Carabao Cup semi-final first leg on Tuesday.
Manchester United boasted two changes from the side that beat Newcastle 2-0 last time out in the Premier League as Ashley Young and Jesse Lingard replaced Antonio Valencia and Juan Mata.
Following a cagey opening, Spurs had the first opportunity to break the deadlock in the ninth minute when Heung-min Son was played in by Christian Eriksen down the left flank and he cut inside before crossing to Harry Winks who fired wide of the far post.
United responded with a half-chance two minutes later when Young’s cross was inadvertently flicked on towards the back post by Toby Alderweireld to Lingard, but he sent a right-foot effort high and wide of the target.
On 17 minutes Rashford was sent through by Ander Herrera and then dinked the ball past the onrushing Lloris only to watch as it dropped the wrong side of the post. However, the United hitman was let off the hook by an offside flag up on the far side.
A minute later Anthony Martial burst into the area but watched as Lloris turned his angled effort wide of his front post.
Spurs thought they had got the opening goal just after the half-hour mark when Dele Alli bundled a Kieran Trippier cross towards goal and Harry Kane touched the ball over the line from close range only to see the offside flag raised against him.
Spurs were then disrupted by an injury to Moussa Sissoko, who was forced off on 43 minutes, before United got their noses in front a minute later.
Paul Pogba picked up a Lingard pass in the middle of the pitch before playing a superb through ball to pick out the run of Rashford who advanced into the box and fired his low effort across the drive of Loris and into the far corner of the net.
Kane looked set to get Spurs on level terms in the 48th minute when he was played in by Alli in a similar position to where Rashford opened the scoring for United, but he hit his right-foot effort too close to De Gea who saved with his foot.
Moments later, De Gea was called into action again as he dived to his left to parry away a Alli header after he got up to meet a Trippier cross in the middle of the area.
On 54 minutes Pogba met a Young cross from the right byline and got a header on target, but it was too close to Lloris who managed to gather.
Two minutes later, Pogba charged his way into the Spurs penalty area and managed to get a shot away under pressure from Trippier, but his France international team-mate Lloris managed to tip his effort over the top.
Kane had another opportunity in the 59th minute when he got on the end of Eriksen’s right-wing cross but he sent his header into the ground which made it easier for De Gea to save.
Three minutes later, Pogba was again denied by Lloris when Martial teed him up on the edge of the area but his slightly deflected side-foot effort was superbly parried away.
Spurs stepped up a gear and should have been level on 66 minutes when Alli was sent clear by Kane into the area but yet again De Gea was on hand to deny the Londoners as he saved with his feet.
Eriksen then curled a 25-yard free-kick inches wide of the far corner, before De Gea produced a brilliant reactionary save to prevent Alderweireld from stabbing the ball past him after he met a corner in front of goal on 70 minutes.
On 74 minutes Alli created an opening for himself inside the area, but the inspirational De Gea dived to his left to push his effort wide of the far post.
Kane looked certain to finally get the equaliser Spurs will feel their play deserved with just four minutes remaining when he got the ball on his left foot inside the box, but for the umpteenth time De Gea’s outstretched foot came to United's rescue.