The young Frenchman looked extremely lively in the first period and created the opening goal as he was brought down inside the box, allowing Paul Pogba to score from the rebound after his penalty was saved from Jordan Pickford.
Martial then scored a curling effort to make it 2-0 to United, before Gylfi Sigurdsson converted an Everton penalty to make it 2-1.
But a relieved Mourinho praised Pogba after the game for having the mental strength to take the 27th minute penalty.
"I was lucky", Pogba said after the game. "I admire players who have the courage to take them". Can he improve? I think he can.
"But for me the most important thing is next time we have a penalty he wants to go again".
Martial was named man of the match for his display at Old Trafford as he threatened throughout and scored a sublime goal in the second half to double United's lead.
'It was not a penalty, clearly the player dived. "You are more exposed then because you have to take more risks".
It was a role reversal for United's second as Martial gloriously curled a Pogba pass into the bottom-right corner within four minutes of the restart.
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But second baseman Ian Kinsler's wild throw on a two-out grounder by Yasiel Puig in the 13th let Los Angeles score the tying run. By the time the ninth inning ended, the Red Sox had already used seven pitchers, including David Price on one day of rest.
Pickford denied Rashford with his legs as the England forward got on the end of a through ball from Mata.
Five minutes later another United attack down the Everton right saw Rashford net after Pickford could only parry a shot from Pogba, but the flag was already up for offside.
"We are playing OK", Mourinho said. It was good for us and good for me.
Everton had enjoyed their best spell after going behind to the first United goal with Sigurdsson heading straight at David de Gea from a promising position and Theo Walcott placing two chances too close to the United 'keeper. Some teams without the ball, can resist and keep a clean sheet all the time. "I've always believed it", said Graeme Souness on Sky Sports after the 2-1 win. It was hard for us.
At the other end of the pitch Marcus Rashford proved just as ineffectual up front as Romelu Lukaku, who was benched for the visit of his former club and took his United drought to nine games after coming on as a substitute.
"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".
"We played well in some moments of the game".
'Even without the goal his performance would be a good performance.