The 18-year-old Zimbabwe-born England youth international has joined the Goodison Park club for an undisclosed fee and will join up with their under-21 side.
Galloway was delighted to announce the news, and told the official Everton website: "It means an awful lot to me coming to such a great club
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."
Galloway became the youngest player in MK Dons' history to feature for the club when, at the age of 15, he made his debut as a second-half substitute in a 6-0 FA Cup victory over Nantwich Town in November 2011.
He went on to make his league debut the following April and in total made 17 appearances for the League One club, scoring once.
MK Dons manager Karl Robinson was disappointed to lose Galloway, but wished him well for the future as he told his club's official 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."
Source : PA
Source: PA