"Last night I thought was a good night," he said at a Houston fundraiser. "I think I could have done better. I will do better, God willing."
Biden also said he felt that he faced fewer attacks than he'd faced in the party's first two presidential primary debates.
"It seems like most people have stopped criticizing me on the debate stage, given what happens to them," Biden said, according to a video clip a donor who was in the room provided to CNN. The donor said the crowd broke out in laughter.
"The Democratic Party has an obligation bigger than to itself now. This is about more than Democrats," Biden told supporters, according to the video clip. "The party has to be bigger than its own interests. We have to unite the country because we're in a position if we don't, no one else is going to fill the world vacuum. If we do, we can make changes that are going to be breathtaking."
Biden struck the upbeat tone the day after the 10 leading Democratic presidential candidates debated at Texas Southern University.
It was the first time Biden had shared the stage with Massachusetts Sen. Elizabeth Warren, and he clashed with Warren and Vermont Sen. Bernie Sanders on health care -- repeatedly pressing his two progressive foes on how they would pay for their favored proposal, "Medicare for All," as he argued for a more limited plan that would build on Obamacare.
Biden also tussled with former Housing and Urban Development Secretary Julin Castro, a low-polling contender who went on the attack, at one time making a thinly veiled swipe at Biden's age by questioning whether he'd forgotten what he'd just said minutes earlier.
On Friday, Biden's campaign said in an email to supporters that Castro had taken a "cheap shot" at the former vice president.
Biden didn't repeat that criticism himself, instead telling reporters that Castro "got his facts wrong."
"I mean look, look -- last night was the closest we came to a debate, OK?" Biden said. "We actually had an open debate on health care, and I felt very good about the debate on health care."
"What I saw last night was fewer personal attacks," he said, adding that he is "getting more and more comfortable with the way the debates are moving."