Spurs had won their three matches across all competitions since Mourinho was appointed as Mauricio Pochettino's replacement last month, but they slipped to defeat after a below-par showing.
Rashford put United ahead and, after a superb individual effort from Dele Alli levelled the scores, the striker converted a penalty to end United's three-game winless run.
Victory moves Ole Gunnar Solskjaer's men above Spurs and into the top six in the table, denying Mourinho - sacked by United a year ago - the chance to be the first Tottenham manager to win his first three Premier League games in charge.
United took the lead in the sixth minute, Rashford profiting from a series of ricochets to beat Paulo Gazzaniga at his near post, the goalkeeper making a poor effort to keep the shot out.
Gazzaniga improved by blocking a Mason Greenwood strike before tipping Rashford's swerving long-range drive on to the crossbar as United continued to dominate.
Spurs equalised out of nowhere before the break, though, Alli's sensational first touch deceiving both Fred and Ashley Young, with the midfielder then slotting in his fourth goal in three games.
Rashford restored United's advantage early in the second half, tucking away a penalty he won himself by being clumsily brought down by Moussa Sissoko.
Fred made a good block to repel a Son Heung-min shot, but it was United who continued to make the running, Gazzaniga denying Daniel James twice in quick succession.
David de Gea was called into action late on but comfortable saves from Serge Aurier, substitute Tanguy Ndombele and Alli ensured United closed out victory.