The 19-year-old signed professional terms with the Gunners last season but did not make a senior appearance for the north Londoners.
The combative midfielder was offered the chance to stay on at the Emirates but opted instead to sign a two-year deal with the Robins as they bid to regain their Championship status following their relegation.
O'Driscoll told the club's official website: "Jordan is a versatile player schooled at Arsenal Football Club.
"He is comfortable in possession, strong physically and a welcome addition to the club as we try to change the dynamic of the squad.
"He is a young player we feel we can develop, and one that has all the desire and hunger to succeed."