Maxwell has penned a two-year contract after impressing on trial at the Eco-Power Stadium.
The 20-year-old turned down a new deal at Scottish giants Glasgow Rangers, whom he joined at the age of 16, in a bid to kick-start his career.
Rovers say they saw off competition from clubs in the Scottish Premiership and the EFL to secure Maxwell’s signature.
The youngster’s arrival ends Gary McSheffrey’s somewhat protracted search for a new left back after Branden Horton and Cameron John were released following the club’s relegation to League Two.
He becomes Rovers’ sixth new addition of the summer transfer window.
“I was in last week doing a bit of training and playing a couple of games with the boys so it’s good to get it done,” said Maxwell.
“It was something I wanted to do straight away. It grabbed me.
“This offer gave me a bit of excitement, a bit of a buzz inside so I’m delighted to get it done.”
McSheffrey said: "He coped and dealt with the games he played during his trial really well. He looked comfortable on the ball, he looked like a good decision-maker and a good defender first and foremost.
"We're pleased to address that position because it has been a priority position for us all summer.
"It was important we didn't rush it. We feel he ticks enough boxes and we're delighted he's here."