The 77-year-old was addressing a virtual concert when he said: “Four more years of George, er, George, er, he - we're going to find ourselves in a position where, if Trump gets elected, we're going to be in a different world.”
Many speculated that Mr Biden was thinking of George W. Bush, who was president between 2001 and 2009, or perhaps George H.W. who was in office for four years until 1993.
Trump fans were quick to jump on the video and claimed Mr Biden’s wife Jill prompted the Democratic hopeful and muttered "Trump" under her breath.
Governor Mike Huckabee tweeted: “Biden thinks he's running against George....Washington. He knew George back then. No wonder they "call a lid" every day at 9am.”
Donald Trump Jr made the same joke hours later, tweeting: “It’s getting beyond ridiculous. This guy is not up for the job. The left is using him to get in their radicals. Did Biden think he was running against George Bush or George Washington?”
Seizing on the slip-up, Mr Trump tweeted: "Joe Biden called me George yesterday. Couldn’t remember my name. Got some help from the anchor to get him through the interview. The Fake News Cartel is working overtime to cover it up!”
Mr Biden and his wife appeared during the star-studded 'I Will Vote Concert' which was headlined by Foo Fighters, the Black Eyed Peas, Jon Bon Jovi and Cher. It garnered around 12,000 views and was live-streamed on YouTube.
It’s getting beyond ridiculous. This guy is not up for the job. The left is using him to get in their radicals.
Did Biden think he was running against George Bush or George Washington? pic.twitter.com/fK1u36Npc7
— Donald Trump Jr. (@DonaldJTrumpJr)
It comes as a new opinion poll shows Mr Biden edging into the lead in Texas, raising hopes among Democrats that the party can turn the state blue for the first time since Jimmy Carter in 1976.
The Biden campaign has been boosted by overwhelming numbers of early voters, with nearly 60 million already casting their ballots, amid polling indicating that more are swinging behind the Democrat than Mr Trump. Over seven million of those have been cast in Texas — nearly 43 per cent of all registered voters in the state.
Mr Biden previously accidentally referred to far-right militia the Proud Boys as ‘the poor boys' as he attacked Mr Trump's record on tackling racism during the second US Presidential debate. He also confused his wife with his sister on stage at a rally in California in June.
“By the way, this is my little sister Valerie!” Biden said while grabbing his wife's right hand. “And I'm Jill's husband,” he went on while reaching for his sister's.
The Trump campaign has in recent months turned up the heat on its claim that Mr Biden's mental capacity is fading, giving him the nickname ‘Sleepy Joe’ at rallies.
But Mr Biden has hit back at concerns, telling reporters in June that he is “constantly tested” by the rigours of the presidential campaign and that he was eager to prove himself at debates.
He later said he was “very willing to let the American public judge my physical, mental — my physical, as well as my mental fitness and to, you know, to make a judgment about who I am.”