Pogba captained United to a 2-1 win over Leicester City in their Premier League opener last Friday, just days after returning to first-team training following the World Cup.
Shaw has claimed that Pogba 'does not need a break', and is hopeful of 'sharing some good memories' with the 25-year-old this season.
"Paul doesn't need a break," Shaw told MUTV. "Honestly, he's a beast! He's unbelievable. He only came back three or four days before the game and it just shows how important he is for the team.
"He was captain and is a very, very big leader in the dressing room, as you'll have seen that in the videos from France [at the World Cup] of course.
"It's nice to have him in the team and around the place – he cheers everyone up and gives good vibes. For me, he's one of the most important players, probably in the Premier League – he's got everything. I'm glad to be able to play with him and I hope we can share some good memories together this season."
After the Leicester match, Pogba suggested that he was not entirely happy at United in comments which are understood to have bemused Red Devils boss Jose Mourinho.
The 20-time English champions will look to move onto six points in the Premier League table when they travel to Brighton & Hove Albion on Sunday afternoon.