The 29-year-old defender has put pen to paper on a two-year deal with Rovers, who this season finished bottom of League Two and were relegated from the Football League.
Sutton was offered a new contract at Mansfield but rejected their offer.
Brabin said: “I’m really pleased to bring Ritchie to the club.
“He’s a no-nonsense type of defender who takes pride in keeping clean sheets. He’s an honest lad, who defends first and foremost and I believe he’ll be an excellent addition to the squad.”
Sutton started his career at Crewe and also had a spell at Port Vale but has spent the past four years at Mansfield, who he helped win the Conference title during the 2012-13 season.
He made 38 appearances for the Stags in League Two this season as Adam Murray’s men narrowly avoided relegation.
Sutton told the club’s official website: “I’m over the moon, I’ve been itching to get here all week and get it sorted. Once I knew of the interest, I was dying to come and speak to the club.
“Obviously, the plan is to get back into the Football League next year, and that’s something I’ve got experience in, so I wanted to come in and do that.”