Rodgers, who has been at Anfield for the past two years, has presided over an upturn in the club’s fortunes and guided the Merseysiders to second place in the recently-completed Premier League season as they just missed out what would have been their first English title since 1990.
The Northern Irishman had a year remaining on his existing agreement with an option for a further 12 months.
But having overseen an improvement from seventh in the Premier League last season to runners-up behind champions Manchester City, Rodgers said he was delighted by the prospect of spending more time at Anfield.
“I am both humbled and privileged to be offered the chance to extend my stay at this great club,” he told liverpoolfc.com.
“I am very happy to build on the foundations we have built over the last two years and now we move on to the next phase which I believe will be as exciting, challenging and rewarding.”
Rodgers added: “I would like to place on record my thanks to the players and staff, who have been absolutely brilliant for me since the day I started at this great football club.
“I am grateful to our owners and senior management team who have supported me and who share my vision of how we can progress on the field.
“Finally, I must thank the supporters who are at the heart of everything we do here.
“I am honoured every day to lead this team and I thank the fans for their continued support as together we strive to bring success back to Liverpool Football Club.”
Liverpool’s US-based owners, Fenway Sports Group, have been delighted by the way in which Rodgers has turned the side into Premier League title contenders without a rash of big-money signings and by how players and fans alike have warmed to the former Swansea boss.
In a joint statement principal owner John Henry and chairman Tom Werner said: “We are very fortunate to have a hugely-talented individual leading our football performance and in whom we place our trust to deliver the vision we share for Liverpool Football Club.
“Brendan is at the heart of what we, as an ownership group, are trying to achieve on the pitch.
“This season has reaffirmed everyone’s belief that we can bring football success to Liverpool and we are all committed to working together to achieve that.
“Players and supporters have made it clear how important Brendan is to our success and so to have him commit to us for the long term is a great boost for everyone going forward.”