Pogba's inclusion from the start was a bit of a surprise considering his lack of pre-season in the aftermath of France's World Cup triumph, and Mourinho even put the captain's armband on the midfielder in the absence of Antonio Valencia.
When the hosts were given a second-minute penalty for handball against Daniel Amartey, who leaned into the bouncing ball with his arm, Pogba pulled rank to take the spot kick ahead of Alexis Sanchez and lift it confidently above Kasper Schmeichel.
The Foxes responded well and the left-wing tandem of Ben Chilwell and Demarai Gray caused United problems. The latter supplied Kelechi Iheanacho on 15 minutes, although the Nigerian was flagged offside before shooting inches wide of the post.
James Maddison and Ricardo Pereira were the two debutants in Leicester's XI and they combined for the best equalising chance of the first half. The playmaker signed from Norwich City was teed up 15 yards out but saw his right-footed shot strongly pushed away by David de Gea.
Shaw fired the Foxes a warning shot on the stroke of the break as he exchanged a one-two with Juan Mata before cutting inside onto his right foot to strike low at Schmeichel.
Marcus Rashford was another World Cup star rushed back into action and he showed signs of rust when he hesitated over a one-on-one chance supplied by Mata on 53 minutes, allowing his England teammate Harry Maguire to quell the danger.
Mata himself came close when he was teed up by Alexis Sanchez but his first-time shot from the edge of the box was deflected wide.
Vardy's introduction from the bench made the Foxes more of a threat and his persistence created a chance for Gray, who got across Bailly but could not generate enough power on his volley to seriously trouble De Gea.
Mourinho brought on his own prolific striker to try to secure victory but Romelu Lukaku squandered his big chance on 78 minutes when Schmeichel was quickly off his line to deflect the Belgian's shot inches over the crossbar.
United did not dwell on Lukaku's miss as Shaw gave them a two-goal cushion minutes later. The full-back ran off the back of Leicester substitute Rachid Ghezzal and hooked Mata's pass across Schmeichel and into the bottom corner for his maiden goal in 141 senior appearances.
Vardy passed up a chance on 91 minutes when he blazed over a volley from 10 yards but he did set up a nervy finish moments later when he knocked in a header from close range, after Pereira's cross had come back off the post.
Schmeichel nearly recreated scenes of his father scoring at Old Trafford 20 years ago when the goalkeeper came up for a last-minute corner and got his head on Chilwell's delivery, only to steer his effort off target.
Claude Puel will hope to get some points on the board when Wolverhampton Wanderers visit the King Power Stadium next weekend, while United travel to Brighton & Hove Albion.
MAN UTD (4-3-3): De Gea; Darmian, Bailly, Lindelof, Shaw; Fred (McTominay, 76'), Pogba (Fellaini, 84'), A. Pereira; Mata, Sanchez, Rashford (Lukaku, 67')
LEICESTER (4-2-3-1): Schmeichel; Amartey (Ghezzal, 63'), Morgan, Maguire, Chilwell; Ndidi, Silva (Iborra, 80'); Ricardo, Maddison (Vardy, 63'), Gray; Iheanacho