The former Swansea City boss, who had initially turned down one of the hottest jobs in English football, replaces Kenny Dalglish after the Scot was sacked in the wake of a disappointing eighth place finish last season.
Rodgers, only the 18th manager in the club's 120-year history, said he would dedicate his life to fight for Liverpool and defend its principles on and off the field.
"We might not be ready for the title now but the process begins today and it's a new cycle," he told a news conference at Anfield.
Liverpool chairman Tom Werner said the 39-year-old was "our first choice and the right choice".
"A forward thinking coach at the forefront of a new generation of managers, he will bring to Liverpool a style of attacking, relentless football," he declared.
"Our common goal is to see Liverpool play among the best teams in Europe and be the best team in England. We believe this appointment today will put us on the path to achieving the goals we all want at Liverpool."
Liverpool's main owner John Henry added in a statement that nobody was expecting an overnight miracle.
"But we firmly believe that the direction the club is heading in will lead to Premier League championships," he added.
Swansea had announced on Wednesday that Rodgers had been offered the Liverpool job subject to compensation being agreed with the Welsh club. That was settled on Thursday.
"I'm blessed to be given this opportunity," said Rodgers, who has a 20-year career in coaching behind him including a stint with Jose Mourinho at Chelsea despite his relative youth.
"This is a long-term [project]. That was important to me, to come in to a project that is going to be over a number of years," he added.
"For me first and foremost it is to defend the principles of this great club which is about offensive and creative football but with tactical discipline."
Rodgers, who turned down an offer to talk to Liverpool earlier this month when the club's owners then held discussions with Wigan Athletic manager Roberto Martinez, made clear he could not have accepted a director of football over him.
"That was something I made clear I couldn't work with," he told reporters. "What you need is an outstanding team. We will form a technical board that will have four or five people that will decide the way forward."
Dutchman Louis van Gaal had been linked in the media with replacing departed sporting director Damien Comolli but Liverpool now look set to move forward without one.
Northern Irishman Rodgers said his initial rejection of Liverpool, five times champions of Europe, had been 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 via the pl
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