Shaw became the most expensive teenager in history when he moved to United from Southampton for an initial fee of B£27million last month.
To succeed Evra, who was also a big influence in the dressing room, seems a tall order for a player who has only just turned 19, but Shaw does not seem fazed at all.
"I don't feel any pressure to come in for Evra," Shaw said on the eve of United's first pre-season game against the Los Angeles Galaxy.
"He has been great for the club for the last 10 years and I hope I can carry that on.
"I will just try to focus on my own game, what I do on the pitch.
"I'm looking forward to trying to impress the new gaffer."
Shaw's arrival at United had been something of a surprise given that he had supported Chelsea, who were also interested in his signature, since he was young.
Most of Shaw's family are Blues fans too, but the left-back, who only made his full Premier League debut in November 2012, is convinced he made the right decision.
"I signed a contract here because I know what the club is capable of. And I know I will improve at the club."
Shaw was speaking alongside fellow new boy Ander Herrera, who signed from Athletic Bilbao earlier this summer.
The Spaniard is also looking forward to the challenge of trying to become a first-team regular at Old Trafford.
"I will miss my friends and my family but I think that I am ready for this step," the midfielder said.
"It's a big step for me but I am happy for this chance and I think that this is the right moment to come to the biggest club in the world."
- Sports & Recreation
- Patrice Evra