The announcement came three hours after the team had drawn 1-1 at Everton in the Merseyside derby on Monday morning (NZT) leaving them with only three wins from their first eight league games this term.
"Although this has been a difficult decision, we believe it provides us with the best opportunity for success on the pitch," the club's owners said.
"Ambition and winning are at the heart of what we want to bring to Liverpool and we believe this change gives us the best opportunity to deliver it.
"The search for a new manager is underway and we hope to make an appointment in a decisive and timely manner."
Liverpool have only beaten Stoke City, Bournemouth and Aston Villa in the league this season and count a 3-0 thrashing at home by West Ham United among their early results.
They also needed a penalty shootout to overcome fourth-tier Carlisle United in the League Cup at Anfield after being held to a 1-1 draw at Anfield after extra time.
Rodgers, who succeeded Liverpool icon Kenny Dalglish as manager in June 2012 and came close to bringing the title back to Anfield for the first time since 1990 in his second season, invested heavily in new players before the latest campaign.
Belgium striker Christian Benteke came in from Villa for £32.5 million, Roberto Firmino arrived from Hoffenheim for 21 million and England fullback Nathaniel Clyne joined from Southampton for 12.5 million.
Liverpool made a positive start to the season but have since been thrashed at Anfield by West Ham and also lost 3-1 to Manchester United, their first half-showing at Old Trafford described as the worst in living memory.
Draws against Girondins Bordeaux and FC Sion in the Europa League and lowly Carlisle - despite the subsequent penalty shootout win - proved to be the final straw.
Rodgers (42) has now become the seventh man to arrive at Anfield and leave without taking the club back to the pinnacle of English soccer which appeared to be their divine right from the mid-1960s to 1990 when they won their last title.
GOOD START
The former Swansea City manager's tenure began so well when, after guiding Liverpool to a seventh-place finish in his first season, he came close to lifting the title 12 months later.
Liverpool put together an 11-match winning sequence from the beginning of February until the end of April 2014, but with the title within their grasp, it literally slipped away.
Liverpool were top when second-placed Chelsea visited Anfield on April 27 but an infamous slip on the turf by skipper Steven Gerrard allowed Demba Ba to give Chelsea the lead and the subsequent 2-0 defeat severely dented their chances.
Worse was to follow the following week when Liverpool were 3-0 up at Crystal Palace with 11 minutes to play before they drew 3-3 to effectively kill off their title dream with Manchester City crowned champions on the last day of the season.
With Uruguay international Luis Suarez sold to Barcelona and strike partner Daniel Sturridge sidelined for most of the campaign, Liverpool lost their cutting edge in attack and last season proved to be another disappointment.
They exited the Champions League after the group stage and Rodgers' handling of Gerrard's departure from Anfield after the midfielder's 17 years at the club was widely criticised.
Any hopes of a return to the Champions League faded in a calamitous end to the season which finished with a 6-1 defeat at Stoke City in Gerrard's final match for the club.
That was their heaviest league defeat since 1963 and that year came back to haunt them last month when West Ham United won 3-0 at Anfield for the first time in the 52 years since then.
Despite Rodgers' assurances last season that gifted England forward Raheem Sterling would not be leaving Anfield, he was sold to Manchester City for £44 million.
The gamble on troublesome Mario Balotelli also failed with the Italy striker shipped out after one erratic season.
Rodgers' failure to get the best out of expensive signings ultimately led to his downfall and leaves the club still looking for the consistency that was their hallmark during the 1970s and 80s when they won the league title 11 times through to 1990.
