Mr Trump's vice-presidential nominee, Mike Pence, defended Mr Trump days after a New York Times report gave ammunition to the Democratic case against Mr Trump by reporting he may not have paid federal taxes for 18 years.
The debate at Longwood University in Farmville, Virginia, was lively from the start as Mr Kaine, the running mate of Democratic presidential nominee Hillary Clinton, honed in on Mr Trump's refusal thus far to reveal his tax records, saying he had broken a promise to do so.
|Kaine and Pence. (Photo: Reuters)
Mr Trump has said they are under federal audit.
"Governor Pence had to give Donald Trump his tax returns to show he is qualified to be vice-president. Donald Trump has to give his tax returns to show he is qualified to be president," Mr Kaine said.
Mr Pence, who has a low-key style compared to Mr Trump's signature bombast, said as a New York real estate developer Mr Trump had created thousands of jobs and had used US tax laws as they were designed to be used.
The two candidates talked over each other so much in a bid to score points that the debate moderator, CBS News' Elaine Quijano, intervened.
"The people at home can't understand either one of you when you speak over each other," she said.
Mr Pence called Russian President Vladimir Putin a "small and bullying leader" and condemned his actions in Syria, taking a harder line than Mr Trump.
Mr Pence's denunciation of Mr Putin for his interference in the Syrian civil war and support for Syrian President Bashar al-Assad was a departure from the frequent praise of Mr Putin by Mr Trump who has called him a better leader than President Obama and said he could work with him.
"The small and bullying leader of Russia is now dictating terms to the United States," Mr Pence said.
"The greatest nation on Earth just withdraws from talks about a ceasefire, while Vladimir Putin puts a missile defence system in Syria."
Debate crucial for both campaigns
Mrs Clinton is seeking to take advantage of Mr Trump's tax report last week to build on her lead in national opinion polls.
Mr Trump needs to rebound from a rocky performance in his first debate with Mrs Clinton last week with their second encounter coming up on Sunday (local time) in St. Louis.
The two vice-presidential candidates wasted no time in launching broadsides against Mr Trump and Mrs Clinton in the opening minutes of their debate.
"Donald Trump always puts himself first," Mr Kaine, a US senator from Virginia, said, pointing out that when Mr Trump began his presidential campaign last year he called "Mexicans rapists and criminals" and had also voiced the "outrageous lie" that Democratic President Barack Obama was not born in the United States.
Mr Pence, the governor of Indiana, shot back at Mr Kaine that he and Mrs Clinton "would know a lot about an insult-driven campaign" and then accused Mrs Clinton, the former US secretary of state, of bungling foreign policy with large sections of the Middle East "literally spinning out of control."
The debate was the only one featuring the vice-presidential contenders and came as Mrs Clinton has edged ahead of Mr Trump in national opinion polls and in some November 8 battleground states where the election is likely to be decided.Reuters/ABCNews