Pogba was named captain by manager Jose Mourinho and took responsibility for the penalty after a handball by Daniel Amartey, slowly shuffling up before beating Kasper Schmeichel.
The World Cup winner with France in Russia is the subject of recent transfer speculation but kept his calm to put United ahead.
Leicester responded well though with new signing James Maddison forcing a fine save out of United keeper David De Gea as the visitors gave Mourinho’s side plenty to think about.
But seven minutes from the end full-back Shaw scored his first goal in professional football, firing home after an exchange with Juan Mata.
Jamie Vardy pulled a goal back for Leicester in stoppage time to give them some reward for a good performance against last season’s runners-up.
In his post-match interview, Manchester United manager Jose Mourinho harped on United’s lull on the transfer market.
“It’s important to win and play well, we played well in periods against a good team that invest more than us.
“We must get used to teams who have players of the same quality that we have. Every team is a good team, forget the name, forget the history, forget the shirt.
“It will be a difficult season for everybody, not just for us. I had my plans for many months and I find myself with the market closed in a situation I did not think I would be in.
“It’s the last time I speak about it. It’s over, the market is closed.
“That’s football management. I think football is changing and managers should be called head coaches.
“We have to wait until the end of November to understand if we can fight for the title,” he said.