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.
American Bradley, 58, 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.
Three members of the Italian`s backroom staff - Diego Bortoluzzi, Gabrielle Ambrosetti, Claudio Bordon - have also left the Welsh club.
Swans chairman Huw Jenkins said he was "disappointed" to sack Guidolin but added that the club "needed to change things as soon as possible in order to move forward in a positive way".
The chairman said Bradley is viewed as a "long-term appointment" who will "stabilise matters on and off the pitch".
"He is highly regarded as a coach and has a wealth of experience on the international and domestic front," added Jenkins.
"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."
Bradley`s first match in charge will be at Arsenal on 15 October, after the international break.
Swansea City Supporters` Trust, which holds a 21% stake in the club, issued a statement saying it is "disappointed" not to be consulted over the managerial change.
"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 wished to work closely with the Supporters` Trust," it added.
"We are also frustrated and angry that the club have allowed the speculation over the manager`s future to be played out in public."
Why Bob Bradley?
Bradley may be something of an unknown quantity to some Premier League followers, but the American`s career is one that Swansea City`s hierarchy have tracked closely.
He is best known for his five years in charge of the United States, with highlights including a run to the 2009 Confederations Cup final that included a victory over then European champions Spain, who were on a 35-match unbeaten run.
Swansea`s American owners, Steve Kaplan and Jason Levien, and chairman Jenkins were particularly impressed by the way Bradley conducted himself during his time in charge of Egypt, dealing sensitively with the 2012 Port Said stadium disaster in which 74 people were killed.
It is believed Swansea were as impressed by Bradley`s character as they were his experience, and he saw off competition from the likes of former Manchester United and Wales captain Giggs to land the Premier League job he has coveted for years.
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