Dele Alli had cancelled out Rashford's opener to leave the two teams level at the interval, but the England forward struck a second from the penalty spot in the 49th minute to secure an important three points.
The victory has seen Ole Gunnar Solskjaer's side move into sixth spot in the Premier League table, while Tottenham have dropped to eighth on what was an unhappy return to Old Trafford for Jose Mourinho.
United started well and made the breakthrough in just the sixth minute of the contest. Indeed, Rashford settled on the ball in a wide position before striking towards the bottom corner, and the Englishman's effort somehow found a route into the back of the net despite the presence of Paulo Gazzaniga.
The strike had pace and direction, but the Tottenham goalkeeper will surely feel that he should have done better as the ball squeezed into the back of the net off the inside of the post.
Harry Kane had the chance to test David de Gea from a free kick in the 12th minute, but the England captain sent the ball high and wide of the target with the visitors struggling to threaten early on.
There was a stoppage in play shortly after when Harry Winks's foul on Daniel James, which brought a yellow card, saw the Wales international take a heavy tumble off the pitch - clashing with Mourinho in the process - but the summer arrival soon returned to the action.
Rashford sent a free kick just wide of the Spurs post in the 18th minute as United, who were full of pace and forward running in the early stages, came close to scoring a second.
Mason Greenwood - on his first Premier League start of the season - then tested Gazzaniga with a close-range strike after James had picked out his teammate with a dangerous cross into the box.
United continued to cause problems and had another chance in the 25th minute when Rashford struck one from outside the box, but the crossbar came to the away team's rescue on this occasion.
Gazzaniga then made a brilliant save to keep out a curling strike from Rashford with Spurs just about holding on.
The Red Devils, inspired by the excellent Rashford, were utterly dominant as the 30-minute mark came and went. Tottenham were having moments in possession of the ball, but the visitors were finding it incredibly difficult to carve out openings at Old Trafford.
The visitors did manage to level the scores in the 39th minute, however, when Alli expertly plucked the ball out of the air before finding the far corner of the net with a smart finish.
VAR had a look at a possible handball, but the goal was given with Spurs equalising despite struggling for much of the first half in Manchester.
United were back ahead in the 49th minute, however, when Rashford, having been brought to the ground by Moussa Sissoko inside the Spurs box, rolled his penalty into the bottom corner of the net.
There was once again a lengthy VAR check, but it was decided that Sissoko had committed an offence inside the box, allowing Solskjaer's team the chance to regain the lead.
Kane then had a low strike blocked behind for a corner with Mourinho's side starting to carry a threat.
United were back on top as the match passed its hour mark, though, and Gazzaniga had to keep out a strike from James during a spell where Rashford was threatening to complete his hat-trick.
Gazzaniga had to make a smart save to keep out a low James strike in the 68th minute, before the visitors made their second change of the match with Tanguy Ndombele being introduced in place of Winks.
United were comfortable as the match headed for its 15 minutes, although there was certainly a sense of nervousness in the crowd after what had happened at the end of the first period.
Luke Shaw was the second and final player off the home side's bench in the final exchanges, and the 20-time English champions closed out the last few minutes to pick up a huge three points.
Tottenham's best chance of a leveller late on fell to Alli in stoppage time, but the Englishman could not find a route past De Gea between the sticks.
Next up for United is the Manchester derby away to Manchester City on Saturday, while Tottenham will attempt to return to winning ways when they host Burnley on the same afternoon.
MANCHESTER UNITED (4-2-3-1): De Gea; Wan-Bissaka, Lindelof, Maguire, Young; McTominay, Fred; James, Lingard (Shaw 87'), Rashford; Greenwood (Pereira 80')
TOTTENHAM HOTSPUR (4-2-3-1): Gazzaniga; Aurier, Alderweireld, Sanchez, Vertonghen; Winks (Ndombele 70'), Sissoko (Lo Celso 86'); Son, Alli, Lucas (Eriksen 64'); Kanebody check tags ::