Hodgson (63) won only seven of 20 Premier League games after taking over from Rafael Benitez in July 2010 and led the club to 12th in the table.
Former player and manager Kenny Dalglish has been placed in charge until the end of the season.
“Both parties thought it in the best interests of the club that he stand down,” said American owner John W Henry in a statement on the club’s website.
Hodgson refused to answer questions about his Anfield future following Liverpool’s 3-1 defeat by Blackburn on Wednesday, the club’s third loss in four Premier League games, which left them just four points clear of the relegation zone.
And when his press conference ahead of Liverpool’s FA Cup third-round match against Manchester United on Sunday was cancelled on Friday, it fuelled further speculation that the former Fulham manager’s tenure was drawing to a close.
“I have found the last few months some of the most challenging of my career,” read a statement from Hodgson on the club’s website.
“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.”
Club ambassador Dalglish will oversee team matters for today’s FA Cup tie at Old Trafford.
The 59-year-old won eight league titles with Liverpool as a player and manager, taking charge of the club for a six-year spell between 1985 and 1991.
He also helped the Anfield side to three European Cup triumphs before hanging up his boots.
“Kenny was not just a legendary footballer, he was the third of our three most successful managers —three giants,” said Henry.
“We are extraordinarily fortunate and grateful that he has decided to step in during the middle of this season.”
There will be sympathy for Hodgson, a proud and decent man who has worked tirelessly but in vain to turn Liverpool around, but brutal reality dictates this was the only decision Fenway Sports Group could take.
Hodgson has extensive domestic, European and international managerial experience, which includes spells with Blackburn Rovers and Switzerland, who he guided to the last 16 of the 1994 World Cup.
He also had a period in charge of Italian giants Inter Milan.
He joined Fulham in December 2007 and steered the club away from the relegation zone, escaping the drop on the final day of that season.
He transformed the Cottagers and guided them to their highest ever top-flight finish in 2008-09, reaching the Europa League final the following year.— BBCSport