The chairman of News Corporation requested the meeting after it emerged that the murdered schoolgirl's mobile phone was hacked by the News of the World newspaper in 2002.
Rebekah Brooks, chief executive of News International, and senior executive Les Hinton both resigned on Friday.
Mr Hinton said he watched the News of the World story unfold "with sorrow".
In a statement following the meeting between Mr Murdoch and the Dowlers, the family's solicitor Mark Lewis said: "We told him that his papers should lead the way in setting the standard of honesty and decency in the field and not what had gone on before.
"At the end of the day actions speak louder than words.
'Deep responsibility'
"He was humbled, shaken and sincere. This was something that had hit him on a personal level. He apologised many times and held his head in his hands."
Mr Murdoch has written an apology for the paper's "serious wrongdoing", which will appear in national newspaper adverts on Saturday.
The advert will state: "We are deeply sorry for the hurt suffered by the individuals affected.
"We regret not acting faster to sort things out."
In other developments:
Former Deputy Prime Minister Lord Prescott called for the resignation of Metropolitan Police Commissioner Sir Paul StephensonEx-News of the World editor Andy Coulson stayed as a guest of Prime Minister David Cameron at Chequers in March, several weeks after his resignation as communications chief, the Press Association quoted a Downing Street source as sayingDowler family solicitor Mark Lewis: "Actions speak louder than words."The meeting between Mr Murdoch and Milly's parents and sister came on the day that Rebekah Brooks resigned amid mounting pressure over the phone hacking allegations.
Ms Brooks was the paper's editor between 2000 and 2003, during which time Milly Dowler's phone was tampered with.
Prime Minister David Cameron said her resignation was "the right decision", his official spokesman said.
In a statement, she said she felt a "deep responsibility for the people we have hurt".
She said she wanted to "reiterate how sorry I am for what we now know to have taken place".
Her statement went on: "I have believed that the right and responsible action has been to lead us through the heat of the crisis. However my desire to remain on the bridge has made me a focal point of the debate.
"This is now detracting attention from all our honest endeavours to fix the problems of the past."
Ms Brooks, 43, who had been with News International for 22 years, bowed to the international pressure piling up on the company.
She has been replaced by Tom Mockridge, who was in charge of News Corporation's Italian broadcasting arm.
Hinton resignation
Later it emerged that Les Hinton, chief executive of the media group's Dow Jones, is also standing down.
Mr Hinton was head of News International from 1995 to 2007, a period in which the News of the World was hacking phones.
In a statement, Mr Hinton, the most senior executive to leave the conglomerate, said he was "ignorant of what apparently happened" but felt it was proper to resign.
James Murdoch praised new chief executive Tom Mockridge as "a highly respected and accomplished media executive", who had shown "leadership and integrity" in creating the Sky Italia 24-hour TV news channel in Italy.
Ms Brooks is still expected to appear alongside Rupert and James Murdoch in front of the Commons media select committee next Tuesday to answer MPs questions on the hacking scandal.
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