The Zimbabwe-born 18-year-old has signed a five-year contract on Merseyside and will join up with the club’s U21 side.
Meanwhile, Sky Sports News understand Everton are also close to completing a season-long loan deal for Chelsea's Ghanaian winger Christian Atsu with personal terms agreed and a medical due to happen this weekend.
Galloway is a current member of England's youth international setup and became MK Dons' youngest ever first-team player when he made his debut against Nantwich Town in 2011 at the age of 15.
The teenager made a total of 17 appearances for the Buckinghamshire club with his only career senior goal coming in the FA Cup first-round win against Halifax Town last season.
"It means an awful lot to me coming to such a great club," Galloway told Everton’s official website. "I'm now looking forward to learning off some of the more senior players - the likes of Sylvain Distin and Phil Jagielka.
"I've been watching them for years now and they're great professionals, great players - players who have been there and done it. That's a big thing for me to be able to learn off guys like that.
"Short-term, I just want to work hard to better myself as a footballer here, which I know I will with the coaches and other staff here.
"Long-term, I also want to get in the Everton team and play week in and week out in the Premier League and play in a team that's doing well, which I know we will do."
MK Dons manager Karl Robinson admitted Galloway’s departure is a blow for the League One side but backed the youngster to be a long-term success at Goodison Park.
“It’s disappointing for the club and it’s disappointing for me to lose a player of Brendan’s quality,” Robinson told the club’s website.
“I do believe Brendan can be as good as he wants to be and I wish him all the luck in the world. He’s going to a fantastic football club in Everton that will only aid his development.”