In his first appearance since winning the 2018 FIFA World Cup with France, Pogba captained Man Utd and took three minutes to score the first PL goal of the campaign, slotting in from the penalty spot after a handball from Daniel Amartey.
David De Gea produced superb saves from James Maddison and Demarai Gray to deny Leicester an equaliser.
Shaw made it 2-0, flicking the ball past Ricardo Pereira and producing a fine finish beyond Kasper Schmeichel with seven minutes remaining.
Jamie Vardy scored from close range in stoppage-time to set up an exciting finish, but Jose Mourinho's side held on for the victory.
Man Utd: 19 Aug v Brighton (A)
Leicester: 18 Aug v Wolves (H)