Liberty Stadium chiefs took the decision to dismiss the Italian on the day he turned 61 after their worst start to a league season in 25 years left them above the relegation zone on goal difference.
Improved performances in the recent defeats to Manchester City and Liverpool were not enough to save Guidolin, with his assistants Diego Bortoluzzi, Gabriele Ambrosetti and Claudio Bordon also being relieved of their duties.
Bradley, 58, had been coaching Le Havre in the French second tier, but met with chairman Huw Jenkins at Bristol Airport last week to discuss a switch to south Wales.
The club's American owners favoured Bradley's appointment given his impressive record with the US national side, while Landon Donovan - who has invested in Swansea and acts as a sounding board for Jason Levien and Steve Kaplan - is believed to have spoken highly of his former boss.
A compensation package has been agreed with Le Havre, and Bradley took charge of the French club for the last time against Sochaux last night. His Swansea reign starts at Arsenal a week on Saturday.
"I have mixed emotions," said Bradley, who narrowly missed out on securing promotion with the French club last season.
"On the one hand I would have liked to carry through Le Havre's project, On the other hand I thank the club for giving me the opportunity to go and work in the Premier League.
"This is a unique opportunity and I will not forget that they have allowed me to leave, or that it is my time here that has showed what I can do and showed I am capable of working in the Premier League."
Manchester United great Ryan Giggs had been the favourite to take over, but it was ultimately felt his lack of frontline managerial experience made him too risky a proposition given Swansea's difficult position.
"We are delighted Bob has agreed to join us," said Jenkins. "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."
The move has not proved popular among many fans, while the Supporters' Trust have angrily accused the club of keeping them in the dark.
The Trust, who hold a 21 per cent stake and have a director on the board, insist they were not consulted at any stage over Guidolin's future.
The row comes just months after the supporters' group also claimed they only became aware of Levien and Kaplan's impending deal to invest in the club through the media.
"Having been an integral part of the club board for 15 years we are saddened that decisions as major as this can be taken without our involvement, despite earlier assurances from the new majority shareholders that they wanted to work closely with the Trust," said a statement.
"We are also frustrated and angry that the club have allowed speculation over the manager's future to be played out in public and want to thank Francesco and his team for their time in Swansea."
Guidolin had only signed a two-year agreement to become Swansea's full-time manager in May, having guided them to survival in an interim capacity last term.
Despite a strong end to last season concerns were expressed over his ability to communicate with his players, as well as the intensity of training sessions.
The Italian's health was also an issue. He missed the games against Arsenal and Norwich last season with a chest infection and was unwell in the lead-up to the defeat against Liverpool.