Rashford scored in each half to give Ole Gunnar Solskjaer, Mourinho’s successor something to cheer.
Dele Alli scored Spur’s only goal in the 39th minute to cancel Rashford’s first goal in the 6th minute. But on return, the gangling striker had the last say via a penalty just four minutes after resumption of play.
With the victory, 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 started brightly against the visitors, playing with energy and adventure and deserved their win.
Smiling manager Solskjaer hailed United’s “best performance for a long time”.
“I thought we dominated the game, could have, should have finished the game there and then (early). Some great saves by their ‘keeper,” he said.
“We’re best when we’re on the front foot,” he added.
“We’re best when we attack quickly, we’re best when we’re aggressive and I thought the boys followed the plan and executed their skills all night.”
Mourinho threw on unsettled midfielder Christian Eriksen in search of an equaliser as the game remained in the balance but despite some late pressure against a nervy United, Spurs left Old Trafford empty-handed.
Mourinho, sacked by United last December, won his first three matches in charge of Spurs, with his team scoring 10 goals in the process but he has problems to sort out at the back as he targets a Champions League spot.
United are also eyeing the top four despite a big gap to close. They remain eight points behind fourth-placed Chelsea and just seven points above the relegation zone.