Manchester United took the lead in the first half of the Premier League clash at Old Trafford when Paul Pogba fired home the rebound after Jordan Pickford had initially saved his penalty in the 27th minute.
And Martial made it 2-0 when he curled home a brilliant first-time finish to put Manchester United in control.
Gylfi Sigurdsson netted a penalty with 12 minutes left to play to set up a tense finale at Old Trafford.
However, the Red Devils hung on to claim all three points and move up to eighth place in the Premier League table.
Martial was in good form throughout, and he missed a glorious chance to put the game to bed late on when his effort was saved by Pickford.
And England legend Lineker clearly approved of what he saw from Martial after he made it 2-0 to the hosts.
Lineker wrote on Twitter during the game: “Terrific finish from Martial. So much talent, so much potential.”
Martial, who has scored four goals in seven Premier League games this season, was named as the man of the match after the game – and he admitted that he is happy with how he has been performing of late.
He told Sky Sports: “I am happy with my form in this moment. I am doing my best to help the team.”
Manchester United will return to Premier League action when they travel to Bournemouth in the lunchtime kick-off on Saturday.
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