Left-back Braham-Barrett, 25, has been on Mark Yates' radar for a while and the Robins manager was delighted to complete a deal.
He told the club's official website: "It's a good signing for us.
"This nearly happened a few weeks ago but we held fire.
"With a bit of prudence and playing around with the money, we realised we had the chance to bring him in.
"We wanted to take a chance and myself and Neil (Howarth) have seen him a few times already. I think the fans will be excited by him because he's direct, he'll bomb on and he will create.
"There were a number of other clubs interested, including a few League One teams, so we are happy to have brought him in."