Mr Trump frequently interrupted, prompting Mr Biden to tell him to “shut up” as the two fought over the pandemic, healthcare and the economy.
The US president was challenged over white supremacist support and refused to condemn a specific far-right group.
Opinion polls suggest Mr Biden has a steady single-digit lead over Mr Trump.
But with 35 days until election day, surveys from several important states show a closer contest.
Overall, the 90-minute debate in Cleveland, Ohio, was light on serious policy discussion. Both candidates talked over each other but Mr Trump cut in some 73 times, according to a count by CBS News.
The tenor became clear early on as the two candidates sparred over healthcare. Hectoring from Mr Trump saw Mr Biden call the president a “clown”.
As they moved on to the Supreme Court, the rancour continued, with Mr Biden refusing to answer when asked if he would try to expand the number of judges.
“Will you shut up, man?” the Democratic candidate snapped at Mr Trump, later adding: “Keep yapping, man.”
Mr Trump, the Republican candidate seeking a second term, responded: “The people understand, Joe. Forty-seven years [in politics], you’ve done nothing. They understand.”
In one of the most talked about exchanges of the night, the president was asked by the moderator, Fox News anchor Chris Wallace, if he was prepared to condemn white supremacists.
He initially said he would but when asked to denounce the far-right Proud Boys group by name, he sidestepped.