Guidolin's 61st birthday quickly turned sour as the Italian learnt he was being forced out after Saturday's 2-1 defeat against Liverpool left Swansea just above the relegation zone with only one win from their seven league matches this season.
Guidolin saved Swansea from relegation after joining the south Wales club in January and was given a two-year contract at the end of last season.
But an American consortium led by Stephen Kaplan and Jason Levien inherited Guidolin when they took over in July and reports began to emerge that his job could be in danger as soon as Swansea began to struggle.
After beating Burnley on the opening day of the season, Swansea went on a six-match winless run in the Premier League, including three consecutive defeats.
That was enough to convince the Americans to ditch former Udinese boss Guidolin and turn to their compatriot Bradley.
"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," Swansea chairman Huw Jenkins said.
"We are obviously disappointed to part company with Francesco.
"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."
Former Manchester United star Ryan Giggs, who left his role as assistant at Old Trafford following Jose Mourinho's pre-season arrival, had been linked with the Swansea job.
But Bradley has left French second-tier side Le Havre to move to the Liberty Stadium, becoming Swansea's fourth manager in less than a year.
Garry Monk was sacked in December 2015 and temporarily replaced by Alan Curtis before Guidolin's appointment.
Bradley led the US to the 2010 World Cup, where his team earned a 1-1 draw with England in the group stages before losing to Ghana in the last 16.
The 58-year-old was dismissed in 2011 after the US lost the CONCACAF Gold Cup final against Mexico and subsequently had a two-year spell as Egypt coach which ended when they failed to qualify for the 2014 World Cup.
Bradley, whose first match will be at Arsenal on October 15, faces a tough task to keep Swansea from dropping into the Championship.
Swansea are above the relegation zone only on goal difference, but Jenkins believes his appointment will prove a shrewd move.
"We are delighted Bob has agreed to join us," he added.
"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."