Two minutes into the game, the hosts were awarded a penalty when Leicester's Daniel Amartey was adjudged to have handled in the box.
Paul Pogba, who was named captain on the day, took responsibility and calmly dispatched the spot-kick into the top corner to hand United the lead.
Demarai Gray was proving to be Leicester's biggest threat down the flank, although the United defence were on their toes to cut out any threat from the winger.
The visitors nearly pulled level on 29 minutes through Adrien Silva's effort from six-yards out, but David de Gea was on hand to keep out the midfielder's effort.
That proved to be the final chance of the half as the Foxes failed to take advantage of their dominance in the closing stages.
Just before the hour mark, CLaude Puel introduced Jamie Vardy and new signing Rachid Ghezzal as they went in search of an equaliser.
Vardy managed to carve out an opening after 75 minutes, after dispossessing Luke Shaw of the ball, he picked out Gray in the box, but the winger was denied by De Gea.
However, United started seeing more of the ball and they should have had a second when Alexis Sanchez played in Romelu Lukaku, who failed to beat Kasper Schmeichel from close range.
Shaw eventually sealed the win with eight minutes remaining with neat finish from a tight angle for his first ever United goal.
Vardy did pull a goal back in the first minute of time added on, but the Red Devils managed to hold on for the win.