The 36-year-old Scot has been confirmed as the successor to Karl Robinson at the Sky Bet League One club after leaving Hearts, and will officially take charge following Saturday's FA Cup second-round match at Charlton.
He will be joined by assistant Stevie Crawford, who also worked alongside him at Tynecastle.
Neilson told MK Dons Player: "I'm delighted
It's great to be here and I'm looking forward to developing the club, developing the players and moving the club forward.
"The most important thing for me was picking the right chairman and the right club and this was the right match at the right time - it's a club that's going places.
"Hopefully the fans are ready for the ride
We want to create excitement but, most importantly, win football matches
With the squad that we've got here, we should be winning football matches - our job is to help them do that."
The Dons, who parted company with Robinson on October 23, and have since had Richie Barker in caretaker charge, are 19th in the League One table
Their 2-1 victory on Saturday at Coventry was the first time they had taken maximum points in nine attempts.
Neilson leaves Hearts with the club lying second in the Scottish Premiership
He guided them to the Championship title in 2014-15 - his first season as a manager - and then a third-placed finish in the top flight last term.
MK Dons chairman Pete Winkelman said: "I'm hugely excited
Robbie has been on our list from day one - he's the most exciting young manager in the UK, by a country mile.
"Hearts are a good club
They didn't stand in Robbie's way and they made it possible.
"They worked with us to make sure it was as smooth as it could be, given that it's in the middle of the season.
"He was at a big club and he was doing well at a big club so we had to get our timing right.
"We wanted to make sure that we would get the right person for the job and I'm so pleased to be sitting here - almost within the time frame that I gave everyone - with the man I believe is going to take this football club forward."
Source : PA