The England forward put United in front early in the first half and converted a second-half penalty to restore their lead after Dele Alli’s exquisite equaliser as Ole Gunnar Solskjaer’s side leapfrogged Spurs, climbing to sixth in the Premier League table.
Spurs went into the match ahead of United after their mini revival under their new boss, who replaced the sacked Mauricio Pochettino at the London club last month.
Struggling United had made their worst start to a season in 31 years but played with energy and adventure and deserved their win.
Smiling manager Solskjaer hailed his youthful side.
“The three points are massive for us,” he told the BBC.
“We’ve had too many draws this season and given too many points away from winning positions. It’s a great lesson the last two games (Sheffield United and Aston Villa) and we came back in a great manner.
“We’ve started the rebuilding. We’ve made decisions that we had to and we’re looking to build this club to be better again and I can’t think short-term when I’m trying to do that.”
The home side deservedly took the lead when Rashford beat Paulo Gazzaniga at his near post in the sixth minute – the forward’s 11th goal in his past 13 games for United and England.
United could easily have stretched their lead but Gazzaniga made amends by touching a thunderous Rashford shot onto the crossbar.