Hodgson had been under huge pressure following poor results this season and the decision came as no surprise with the club 12th in the Barclays Premier League, just four points above the relegation zone.
Kenny Dalglish will take control of team matters for Sunday's third round FA Cup tie with Manchester United at Old Trafford and the rest of the season.
In a statement on the club's website, principal owner John Henry said: "We are grateful for Roy's efforts over the past six months, but both parties thought it in the best interests of the club that he stand down from his position as team manager. We wish him all the best for the future."
Hodgson added: "Being asked to manage Liverpool football club was a great privilege.
"Any manager would be honoured to manage a club with such an incredible history, such embedded tradition and such an amazing set of fans.
"Liverpool is one of the great clubs in world football. I have, however, found the last few months some of the most challenging of my career.
"I am very sad not to have been able to put my stamp on the squad, to be given the time to bring new players into the club in this transfer window and to have been able to be part of the rebuilding process at Liverpool.
"The club has some great, world-class players, with whom it has been a pleasure to work and I wish the entire squad well for the rest of the season.
"I thank those with whom I have built up a close working relationship at the club for their loyalty and support during very testing times, and finally of course to the Liverpool fans, your passion and dedication to the club will see Liverpool at the top of the game once more."
Source: PA
Source: PA