His remarks come as he considers facing off against the former secretary of state in the 2016 presidential race, and they seem to signal that he sees his earlier stance on the raid as a potential liability.
At an event honoring former Vice President Walter Mondale, Biden said he had privately advised the President to pursue the raid on bin Laden's compound after initially advising a more cautious approach at a Cabinet meeting.
"We walked out of the room and walked upstairs," Biden said. "I told him my opinion: I thought he should go, but to follow his own instincts."
The new account is a significant departure from what he said at a Democratic retreat in January 2012.
"Mr. President, my suggestion is, 'Don't go,'" Biden said, according to an ABC News report from that time.
"'We have to do two more things to see if he's there,'" Biden recalled, though the story did not include what those two things were.
The Cabinet meeting Biden referred to has been described by several people in the administration, including President Barack Obama himself.
The President asked his closest advisers for input on how he should respond to intelligence that bin Laden was holed up in a compound in Abbottabad, Pakistan -- intelligence that was anything but certain.
The proposed raid by Navy SEALS was risky, particularly without notifying Pakistan of the plan.
"Those decisions are not always popular. Those decisions generally are not poll-tested," Obama told moderator Bob Schieffer in a 2012 presidential debate. "And even some in my own party, including my current vice president, had the same critique" about the risk of the operation as did some outsiders.
Clinton has also characterized Biden has having been openly skeptical in the meeting.
Tuesday's account is also a change from an account Biden gave on NBC's "Meet the Press" in May 2012, in which he described advising the President to follow his instincts but didn't explicitly advise him to "go" for it.
"We walked up toward the residence, toward his office," said Biden of his conversation with Obama, "and I knew he was going to go (for the raid). And what I always tell him when he -- he looked at me again, and I said, 'Follow your instincts, Mr. President. Your instincts have been close to unerring; follow your instincts.'"
Biden also offered a reason for the difference between what he said in the Cabinet meeting and his new account of the private conversation with Obama in which he advocated following his instincts.
The vice president now says that he suggested the administration undertake additional drone surveillance on the compound while in that Cabinet meeting, but that he made that recommendation because he didn't want to undermine the President if he ended up choosing a more cautious approach.
"Imagine if I had said, in front of everyone, don't go or go and his decision was a different decision," said Biden. "It undercuts that relationship. So I never, on a difficult issue, never say what I think finally until I go up to the Oval with him alone."
Biden also said Tuesday that only two people in the meeting were definitive in their advice to the President, contradicting Clinton's account of how she supported the mission.
There were "only two people who were definitive and were absolutely certain," he said, referring to the men who were, respectively, the director of the CIA and the secretary of defense at the time. "Leon Panetta said, 'Go,' and Bob Gates -- who has already publicly said this -- said, 'Don't go.' And others were at 59/41."
In her 2014 book "Hard Choices," Clinton writes that she was an immediate supporter while Biden "remained skeptical."
"I respected Bob [Gates] and Joe [Biden]'s concerns about the risks of a raid, but I came to the conclusion that the intelligence was convincing and the risks were outweighed by the benefits of success," she wrote. "We just had to make sure it worked."
Biden also drew attention in his remarks Tuesday to the fact that he knew about the intelligence on bin Laden's location before Clinton did.
"The President and I, and only two others in the administration, knew about Abbottabad as early as August" 2010, Biden said Tuesday. "We did not go for almost a year to get him. And major players in the Cabinet did not know about it till January or February (2011)."
In Clinton's account, she learned about the intelligence in March 2011.