The minnows had sacked Rob Edwards earlier, after a poor run of results following relegation to the Championship last season.
Having expected to fight for promotion, Luton will likely have to wait a year for a dream top flight return.
Town CEO Gary Sweet said: “It is with great pleasure and excitement that we are able to announce Matt’s appointment on a long-term contract.
“Before anything else, I would like to thank Wycombe for their co-operation in ensuring a swift and smooth transition. It may be flattering to be on the receiving end, it’s never pleasant, but we have made sure that we’ve gone above and beyond to compensate Wycombe to secure his services, along with his trusted coaching team.
“Over the years in which we as a club have risen up the EFL, we encountered Matt at every step as a respected captain of his team, and together they have always shared our hunger and desire to improve and get to the next level.
“Since returning to Wycombe as manager we have been hugely impressed with his work in leading them back towards the top of League One, and especially in how they have done it in such an entertaining way, forging a reputation for front-footed attacking football.
“We identified Matt’s personal and team-building qualities as the outstanding candidate to fit in with the culture we have long established here at Luton Town, and I’m sure every supporter will join me in welcoming him, along with Richard, Lee and Ben to Kenilworth Road.
“Finally, I’d just like to thank the immense amount of work and attention to detail our recruitment team have put into identifying the best candidate and impressing him in meetings, along with the fantastic, as ever, and unanimous support of the board in arriving at a quick decision after interviewing Matt.”