Manchester United great Giggs had been the favourite for the role, while Paul Clement, Patrick Vieira and Jurgen Klinsmann were also considered.
But it was felt a lack of frontline managerial experience counted against the Welshman 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.
“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.”