Dalglish's departure comes after face-to-face talks with principal owners John W Henry and Tom Werner in Boston on Monday.
The 61-year-old has paid the price for Liverpool's poor Premier League performances, especially at Anfield.
"I am disappointed with results in the league, but I would not have swapped the Carling Cup win for anything as I know how much it meant to our fans and the club to be back winning trophies," said Dalglish.
"It has been an honour and a privilege to have had the chance to come back to Liverpool as manager.
Dalglish announces resignation in 1991
"Whilst I am obviously disappointed to be leaving, the matter has been handled by the owners and all concerned in an honourable, respectful and dignified way and reflects on the quality of the people involved and their continued desire to move the football club forward."
Werner insisted that the club owed a "great deal of gratitude" to Dalglish.
"Kenny came into the club as manager at our request at a time when Liverpool Football Club really needed him," he added.
"He didn't ask to be manager; he was asked to assume the role. He did so because he knew the club needed him."
Dalglish had been expected to take a holiday this week, but flew to the United States instead to review the season with Henry and Werner.
Former managing director Christian Purslow told BBC Radio 5 live that he does not think that should have been a factor.
"I think it's unfair and unrealistic to expect a football manager to be front and centre on an issue of that kind," he said. "I would never expect my manager to make decisions about communications policy, and a sensible legal matter.
"He should not have been put in that situation. He does not have the training you might need to handle that situation. To judge him for that, I would be astonished if that would play any part in the decision to remove him as manager and if so I would take issue with that."
Suarez has been the most successful of Dalglish's signings on the pitch, but critics have questioned whether the England trio of Andy Carroll, Jordan Henderson and Stewart Downing represented value for money at a combined cost of approximately £70m.
But Purslow does not think Dalglish should have been sacked: "I'm extremely surprised by Kenny leaving and personally very disappointed. I suspect he's very upset about leaving the club.
Mike InghamBBC radio 5 live chief football correspondent
"It's not entirely a surprise. The writing was starting to appear on the wall. I think Dalglish has paid a price for the Luis Suarez affair. The thing about Dalglish is that he is so supportive of his players and in the end I think he took the You'll Never Walk Alone thing too far."
"They invested a lot of money in his team and I'm not sure they've given him anywhere near enough time to make the investment work. It seems bizarre to me to have a situation where you appoint someone for three years and make significant investment and, one year in, lose faith in that.
"Kenny came into the club initially to help me and the board behind the scenes and then stepped in at a terribly difficult time in the club's history. The previous manager had been starved of funds, the squad desperately needed investment and he did that.
"Everybody would say he unified the club and made Liverpool a football story again. He made us an attractive team to watch again."
Dalglish achieved legendary status on Merseyside during his own time as a Liverpool player, making 515 appearances and scoring 172 goals after signing from Celtic in August 1977.
He became player-manager in May 1985, collecting three league titles and leading the team's commemorations of the 96 supporters
who died in the Hillsborough Stadium disaster,
before announcing his shock resignation in February 1991.
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