The central midfielder has signed a three-year contract with the Whites.
United beat off interest from a number of clubs, including Blackburn Rovers and Wolves, to capture the signature of the highly-rated 23-year-old.
Brian McDermott told the official club website: “Myself, the owners, the staff, the players, and I’m sure and I can speak for the supporters on this one, are absolutely delighted to sign Luke.
“He’s a player I’ve known for a long time, I’ve watched his career progress, and I feel he was the best player in his division last year.
“Luke certainly fits the profile of player that we’re looking to attract: He is young, hungry, wants to work hard and very obviously talented.
“I’d like to thank the owners for making this deal happen. We had spoken about Luke earlier in the summer being one for the key signings we’re looking to make.”
Newly-appointed chairman Salah Nooruddin said: “We’re delighted to complete the deal for Luke. There was plenty of interest in him from other clubs, but Brian had never made any secret about his desire to bring Luke here, and we’re pleased we were able to get things done and welcome a very good addition to our squad.”
Murphy is a graduate of Crewe’s Academy system and, since making his debut as an 18-year-old, he made 186 appearances for the club, scoring 25 goals.
Managing director David Haigh said: “It’s great to welcome Luke aboard. He is a player that Brian had identified some time ago and we have all worked very hard to make this happen.”
Murphy becomes the club’s second signing of the summer following a deal to bring in striker Matt Smith from Oldham Athletic.