Brendan Rodgers from Swansea (Photo: Reuters)
Rodgers, who had initially turned down one of the most demanding jobs in the Premier League, replaces Kenny Dalglish after the Scot was sacked in the wake of Liverpool's disappointing eighth place finish last season.
"He was our first choice and the right choice," Liverpool chairman Tom Werner told a news conference at Anfield.
Liverpool's main owner John Henry added in a statement: "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."
Swansea had announced on Wednesday that the 39-year-old Rodgers had been offered the Liverpool job subject to compensation being agreed with the Welsh club. That was settled on Thursday evening.
"I'm blessed to be given this opportunity," Rodgers said.
"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."
Rodgers turned down an offer to talk to Liverpool earlier this month and the Merseyside club's American owners then held discussions with Wigan Athletic manager Roberto Martinez.
The Northern Irishman said Liverpool, five times champions of Europe in their heyday, had been too good an opportunity to turn down. He said his initial rejection was out of respect for his previous employers.
Rodgers 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 (second division) via the playoffs after moving to Swansea in 2010 following spells in charge of Watford and Reading.
Liverpool have been searching for a replacement for Dalglish, whose second spell in charge at Anfield ended after their worst league finish for almost 20 years.
Dalglish did steer Liverpool to a League Cup triumph and an FA Cup final appearance last season but the club are desperate to get back into the top four and mount a title challenge.
"Brendan Rodgers's 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." Henry said.
"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."
Once the dominant force in English football, Liverpool have not won the championship since 1990.
They finished the 2011-12 campaign 37 points behind champions Manchester City and recorded their fewest wins (14) in a top flight season since 1953-54.
"The style of football Brendan is associated with is exactly what we want to see at Anfield," Werner added.
"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."
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