Guidolin's future has been in doubt after a poor start to the season which continued on Saturday with a 2-1 home defeat by Liverpool that left Swansea fourth bottom of the table.
The American-owned club announced on their website that Guidolin, who joined in January following the sacking of Garry Monk, had left and that the former U.S. national coach would take over.
Bradley will take charge of French club Le Havre for the final time on Monday when they take on Sochaux.
"We are obviously disappointed to part company with Francesco," Swansea 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."
Bradley is a vastly-experienced coach who managed in the MLS with Chicago Fire and MetroStars before becoming U.S. coach from 2006-11. He also had a spell in charge of Egypt.
"We are delighted Bob has agreed to join us," Jenkins said.
"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."
Bradley faces a tough start once the Premier League season resumes after the international break with his first match in charge away to Arsenal.
His son Michael is currently the U.S. captain and has played in the Bundesliga with Borussia Moenchengladbach and with Roma in Serie A.
American businessmen Jason Levien and Steve Kaplan brought a controlling stake in Swansea in June.
(Reporting by Martyn Herman, editing by Ed Osmond)
