Shaw left Southampton for Old Trafford earlier this summer for an initial transfer fee of £27million, while Evra departed for Juventus last week.
Frenchman Evra was United’s first-choice left-back for eight years during which time he won five Premier League titles and played in three Champions League finals.
Shaw, currently on tour in the United States, said: "I don't feel any pressure to come in for Evra.
"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.
“And hopefully I can do that in these pre-season games.”
The 19-year-old was a Chelsea fan growing up, but admits the allure of playing for United won him over in the end.
"Manchester United is the biggest club and it will push me to do my best," added Shaw.
"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."