The Swans have not won in the Premier League since the opening day of the season and are 17th in the table.
Bradley, who has also managed Egypt and Norwegian side Stabaek, leaves French second-tier team Le Havre to succeed Guidolin, who was appointed in January.
The club's hierarchy spoke to several potential candidates, including former Wales captain Ryan Giggs.
Bradley will take over a Swansea side who have lost their past three league matches and find themselves above the relegation zone only on goal difference.
Guidolin has been under intense pressure - which was increased by Saturday's 2-1 home defeat by Liverpool - and his sacking was announced on his 61st birthday on Monday.
Swans owners Steve Kaplan and Jason Levien and chairman Huw Jenkins were impressed by Bradley's experience and character after meeting the 58-year-old.
His first match in charge will be at Arsenal on 15 October, after the international break.
Swansea have confirmed that three members of Guidolin's backroom staff, Diego Bortoluzzi, Gabrielle Ambrosetti, Claudio Bordon have also left the club.
"We are obviously disappointed to part company with Francesco," chairman Huw Jenkins said.
"We felt he deserved his opportunity after the work he did last season. Unfortunately we have not been able to carry performances over from last season and we felt we needed to change things as soon as possible in order to move forward in a positive way.
"We would like to thank Francesco and his staff for their services to the club and wish them well for the future.''
Jenkins said Bradley is a long-term appointment.
"We are delighted Bob has agreed to join us," he said.
"He is highly regarded as a coach and has a wealth of experience on the international and domestic front.
"He is well aware of the club's footballing philosophy and will provide us with strong leadership qualities and a renewed belief to compete at this level.
"It is never easy changing managers, but we are looking at a long-term appointment and we are confident Bob can settle us down and stabilise matters on and off the pitch."