The result also vindicated Mourinho's decision to drop struggling centre-forward Romelu Lukaku from his starting line-up after the Belgian worldwide had failed to score in his previous eight games.
"We played well in some moments of the game".
Gylfi Sigurdsson's late penalty ensured an anxious end to proceedings for the hosts but, with United's legendary former star Eric Cantona watching from the stands, his fellow countrymen did enough to secure three points.
Everton carved out another good chance when Lookman found Richarlison and he played Coleman into space for a rising shot that cleared the crossbar when he really should have made De Gea work.
"We have on us that extra factor that we are Manchester United, but we have to be strong enough and courageous enough to know that it's a game that we can win". But a lack of first team appearances under boss Jose Mourinho may mean he could look elsewhere.
Martial doubled the home side's lead in second half with a stunning strike from the edge of the box, which has now become a trademark Martial finish.
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"This victory was very important to get us back toward the top of the league".
"We are playing OK", Mourinho said.
The Frenchman has managed to score five goals in all competitions for Manchester United so far this season.
'We were unlucky that we didn't win against Chelsea in the last second and, as you saw, in this game also in the last minute we had some trouble after a corner kick. For us it is nearly an impossible mission to keep a clean sheet even with the phenomenal goalkeeper [David de Gea] we have and to score goals is really hard for us. Even if he doesn't score, to create what he is creating, it is a solid performance for the team.
"It took quite a long time for his brain but also his body to be ready to play the way we want him to play".
Unfortunately they soon took their foot off the gas and let Everton back into the match which forced the players to defend for their lives. "He's got that good of a penalty, he's got that power in it, so you've got to go a tad early".