The 22-year-old had signalled his intentions to leave Fir Park by rejecting a new contract with the Scottish Premier League side last month, ending a lengthy association with the Steelmen, whom he joined 12 years ago as part of the academy.
With a number of clubs interested in acquiring the former Scotland U21 international, it was Jones and Luton who won the race for his signature, with the Town boss saying: "We are delighted to get Allan done as he’s one we’ve been tracking for a while.
“He’s one of the best young midfield players in Scotland, athletic and hungry for success.
“He’s one that I’m sure will be a fans’ favourite at Kenilworth Road.
“From the moment I met him I knew he was serious about his football, and we like individuals like that.
“He had a lot of interest, so we’re delighted he’s chosen to join us.
“He’s already got vast experience, but at 22 years old he’s a great age and one that can grow with us.
“We couldn’t have done this without the support of the season ticket holders who have donated their credit.
“As with the signing of Fred, we thank everyone for that wonderful support in helping us pay the fee to bring him south.”
The Steelmen, managed by former Town defender Graham Alexander, had fully expected Campbell to move away this summer following negotiations over a new deal breaking down recently.
Speaking to the BBC, Alexander had said: "We did offer him a very good contract for us a couple of weeks ago.
"I didn't want to get to summer and he goes elsewhere and we hadn't tried everything we can to keep him."