The England Under-21 centre-half had a medical with United on 8 June before a late hitch delayed the transfer.
And despite offers from Arsenal and Liverpool, the 19-year-old insisted he only wanted a move to Old Trafford.
"It's a once in a lifetime opportunity and I'm looking forward to a new challenge," Jones said on Twitter.
"I'd just like to say that I've loved my time at Blackburn and I will always be a huge fan."
A United statement said the club were "delighted" to conclude the transfer.
"The player passed a medical in Manchester last week and has agreed a five-year contract," the statement read.
"The deal will be completed once the player returns from international duty."
Jones only made his Premier League debut in March last year but has made a big impact in his first full season and featured for the England U21s in the 1-1 draw with their Spanish counterparts on Sunday.
Several United players expressed delight on their Twitter accounts on Monday afternoon after hearing about their new team-mate.
Striker Wayne Rooney said: "Welcome to Man Utd Phil Jones. Looking forward to playing with u," while defender Rio Ferdinand said: "Welcome to Phil Jones. Another top young talent joining the club. The future is bright."
Blackburn boss Steve Kean revealed the club had made a "staggering" offer in an attempt to keep hold of their star defender.
"Phil wants to play Champions League football now and we're not at that level," Kean said.
"I'm sure we will be in the coming years, and we gave it a real good try and made him a fantastic offer, but I understand why he wants to go now.
"He goes with our blessing and it's an opportunity for us to invest in other young players across other areas of the pitch.
"We were hoping he was going to be captain of our club and captain of England, he's a fantastic prospect and I'm sure he'll be an England regular very soon."
The teenager will face a battle to immediately break into United's first team with Ferdinand and Nemanja Vidic the clear first choice for the centre-half spots.
Chris Smalling and Jonny Evans are also competing for places at the heart of Sir Alex Ferguson's defence.
A string of fine performances by Jones helped Blackburn end their season with a four-game unbeaten run, which included a 1-1 draw with United and wins over Bolton and Wolves, to guide them to Premier League survival.
Preston-born Jones joined Blackburn's youth set-up in 2002, making his debut in a Carling Cup win at Nottingham Forest in September 2009. His first Premier League appearance came six months later in a 1-1 draw with Chelsea at Ewood Park.
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