Shaw faces Ashley Young for competition at left-back with the 33-year-old working his way back to full fitness after World Cup duty with England but will hope to start once again when Jose Mourinho's men face Brighton in their next league game next Sunday (4pm).
A Shaw giveaway deep in his own area in the 75th minute nearly gifted Leicester the leveller, but De Gea made another spectacular save to stop Demarai Gray's first-time shot from a pinpoint Vardy cross.
Pogba has been linked with a move away from Old Trafford this week after stories emerged that the Frenchman has grown unsettled by some of Mourinho's comments during the summer.
Shaw wasn't positive his goal meant he'd remain a first-choice starter for Mourinho, but he said he's set his sights high. "We thought maximum (he could play) 60 minutes, but he managed 80".
Asked if he had the tools to compete without new signings, Mourinho said: "I have my players, I like my players, I like to work with my players". But, of course, in the second half, I had Andreas Pereira and Fred both with a yellow card. We were in trouble.
I know they had a lapse right at the end when Jamie Vardy pulled a goal back for the Foxes but, otherwise, they had the kind of focus they needed. The Portuguese manager was pointedly critical of his star player following the World Cup, essentially challenging him to replicate his World Cup form at the club level.
"We don't have a penalty taker, it's how we feel and I made a decision to take it".
After the United goal though it was mostly one-way traffic for the visiting side, with David De Gea repeatedly called into action including a miraculous save on a James Maddison shot that should have drawn Leicester level.
Shaw, who himself has suffered plenty of criticism from Mourinho in the past two seasons, then appeared to make the game safe when he fired low into the far corner to spark wild celebrations from his manager.