The midfielder, who has been linked with Hibs and Aberdeen in the past, has opted for a switch to the English Championship side.
Campbell has joined after more than 150 games for Motherwell.
The Fir Park side confirmed they “reached a financial arrangement with Luton based on our training compensation entitlement. We have also negotiated the addition of future economic rights”.
Luton Town boss Nathan Jones labelled him “one of the best young midfield players in Scotland” and believes he will become a fans' favourite at the club.
"It has a really good feel to it,” the midfielder told the Luton website. “I have heard a lot of good things about the club and how close they are with the fans.
"I can’t thank the Motherwell supporters enough for what they did for me. But as a footballer, there is a time in your career where you feel you need to take that next step and Luton have given me the chance to come down here and really try and prove myself at this level, kick-on and develop me as a player and a person.
"I think this is the next level up. I am an ambitious person, I set targets for myself and I’ve come down here, set targets and now I am going to try and get them.”
He added: "I spoke to the manager, Mick [Harford] as well, and they really want to try and develop me as a player and that is the main thing.”