Martial was introduced with just over 20 minutes to go, and United's attacking instincts instantly sharpened. But it was the hosts who almost scored next, via a superb curling free-kick from the left-foot of Rachid Ghezzal. Fortunately for Solskjaer's men, De Gea was able tip the ball over before it found the top corner.
Romelu Lukaku was denied by Schmeichel as the Reds searched for a second in the final 10 minutes, before Vardy shot tamely at De Gea at the end of a lively Leicester counter.
Martial was then fed by Young in the closing minutes of normal time, but like his Belgian colleague, shot straight at the keeper. Leicester sought to pile on the pressure as the four minutes of stoppage time ebbed away, but Maguire could only clip his shot softly to the left of De Gea's upright.
The whistle blew, the Leicester fans booed, and the United away cheered buoyantly. It was a ninth win in ten for Solskjaer, and three more priceless, hard-fought points earned on the road.
United: De Gea; Young (c), Lindelof, Bailly, Shaw; Matic, Pogba, Herrera; Lingard (Jones 90), Alexis (Martial 67), Rashford (Lukaku 77).
Subs not used: Romero, Mata, Fred, Dalot.
Scorer: Rashford 9
Bookings: Lingard, Matic, Shaw, Martial
Leicester: Schmeichel; Pereira, Maguire, Evans, Chilwell; Ndidi, Mendy (Iheanacho 84), Maddison (Ghezzal 62); Gray (Okazaki 74), Barnes, Vardy.
Subs not used: Ward, Morgan, Fuchs, Choudhury.
Bookings: Vardy, Mendy, Ghezzal, Ndidi