Leicester lost former United midfielder Danny Drinkwater to a red card late in the piece but it did nothing to alter the result as they hone in on a their first-ever top flight premiership.
French forward Martial was picked out on the left side of the area by Antonio Valencia before producing a quality finish beneath Kasper Schmeichel in just the eighth minute to give the Red Devils a dream start.
Just over five minutes later, Schmeichel came up with a vital save to deny Jesse Lingard's angled effort after a dominant beginning, before the Foxes struck back in the 17th minute.
Drinkwater sent a floating ball into the penalty area which allowed Morgan to rise above his marker and glance home a header past David De Gea.
There were a couple of controversial moments in the first half, namely when Danny Simpson gave the ball away in midfield but fortunately managed to track back and out-muscle Lingard, just as Schmeichel advanced to clear.
There were calls for a red card for the Leicester right-back because of his tackle as the last man but referee Michael Oliver waved away the protests.
Leicester were then unlucky not to have a penalty when Riyad Mahrez produced his customary trickery when he checked back on Marcos Rojo, who stuck a leg out and felled the Algerian winger in the box, but nothing was doing from the match official.
The second half was slightly more dour between the two sides with only a Chris Smalling header, which was smartly kept out by Schmeichel, the best chance of note.
The Dane was later called on to fist away Wayne Rooney's free-kick after Drinkwater had been given his marching orders for a second bookable offence for bringing down Memphis Depay just outside the area, which was the last action of real consequence.
Anything other than a Spurs win will see the Foxes clinch the trophy which they could proudly lift at the King Power Stadium after the Everton fixture next weekend.
2015/16 Barclays Premier League - season scores