"If they're going to take credit for it, fair enough, for what they've done while they're there. But it ought to go with a healthy dose of 'Thank you, George Bush' up front and a recognition that some of their early recommendations, with respect to prosecuting that war, we're just dead wrong," he said.
Biden countered that it's not credit the Obama administration is taking, but responsibility.
"We're not taking credit. We had to take responsibility," said Biden, who is in Vancouver for the Winter Olympic games. "When we took responsibility for the mess that we were handed to us at the end of last year, an awful lot of very informed news people ... were wondering whether or not that country could be put together."
President George W. Bush has been mostly silent since leaving office more than a year ago, while his vice president appears to be relishing the freedom that comes from being out of office.
"I have the great freedom and luxury of speaking out, saying what I want to say, what I believe," Cheney said.
Some analysts point to a power vacuum in the Republican Party that has left a wide opening for the former vice president to say his piece. Cheney may also be looking for some public vindication.
"Since they lost in 2008 and until they find a presidential candidate in 2012, leadership is up for grabs, and he's grabbing it," said ABC political analyst George Will.
On Sunday, the former vice president pointed to several Bush policies that the Obama administration has not been able to fully roll back, including the operation of the detention facility at Guantanamo Bay and the use of military tribunals.
Despite the differences in the Sunday talk show duels, there were some points of agreement. Cheney said he supports efforts to end the "Don't Ask, Don't Tell" policy in the military and the Obama administration's approach to the war in Afghanistan.
"When the [military] chiefs come forward and say, 'We think we can do it,' then it strikes me that it's time to reconsider the policy," Cheney said.
In Afghanistan, Obama approved a surge of 30,000 additional troops last year and his administration has increased drone attacks in neighboring Pakistan, where some al Qaeda leaders are thought to be hiding. At the same time, Obama set July 2011 as the date of the start of withdrawal of U.S. troops from Afghanistan.
"You know, I'm a complete supporter of what they're doing in Afghanistan. I think the president made the right decision to send troops into Afghanistan. I thought it took him a while to get there," Cheney said.