The Merseyside club presented Northern Irishman Rodgers as the successor to Kenny Dalglish at a press conference on Friday.
Rodgers said: "I'm blessed to be given this opportunity. I want to thank John Henry, Tom Werner and FSG for the opportunity to manage such a great club.
"I'm really excited and I can't wait to get started on this incredible project going forward. I promise to dedicate my life to fight for this club and defend the great principles of Liverpool Football Club on and off the field."
Liverpool's Principal Owner John Henry was adamant that the club got the man they wanted.
"Fenway Sports Group is committed to building for the long-term," he said. "Brendan Rodgers' appointment today as manager of Liverpool Football Club is one of the most important steps we will take in building the kind of club on and off the pitch supporters can be excited about.
"Brendan's comprehensive football philosophy is perfectly aligned with those at the club and those soon to join the club. He was the first choice unanimously among them and he had no hesitation at all in embracing exactly what we want to try to build at Liverpool.
"We do not expect miracles overnight nor should anyone else. But we firmly believe that the direction the club is heading in will lead to Premier League championships. We will embrace the unconventional, build the right way and together set a bold, exciting course for this historic club."
Chairman Tom Werner added: "The style of football Brendan is associated with is exactly what we want to see at Anfield. Aggressive, attacking play; Swansea were a revelation last season with their brand of entertaining football.
"But it's not just what he has achieved in the last two years that has impressed us; he has dedicated his career to coaching, has studied the game all over Europe and he is the perfect person to take this club forward."
The 39-year-old guided Swansea to a respectable 11th place finish in the Premier League last season, their first campaign at the highest level since 1983.
He led the club to promotion from the Championship via the play-offs after moving to Swansea in 2010 following spells in charge of Watford and Reading.
Rodgers said that he would not be prepared to work with a sporting director at Liverpool, but the club did confirm that they would be looking to develop a football operations structure to support the manager, with further details to be announced in the future.
Swansea chairman Huw Jenkins has confirmed that the departing Rodgers will not be able to sign Swansea players for the next year, as part of an agreement.
"We have got a good relationship, we have dealt with it in the right way with Brendan and Liverpool," Jenkins told Talksport. "We have got some protection on Brendan coming back for our players in the initial period, which I think is the right thing.
"He has got a timescale to assess things there and that is going to take him while so I’ve got no worries there. We have got a 12 month respite."