Trump, a real estate tycoon making his first run for public office, said Clinton's long years of service represented "bad experience" with few results and suggested her disavowal of a trade deal with Asian countries was insincere.
For Trump, 70, the debate was a chance to appear disciplined. For Clinton, 68, it was an opportunity to reassure voters she could be trusted. It remained to be seen how voters would judge their performance.
In a sign investors saw Clinton as the winner, Asian shares recouped early losses on Tuesday and the dollar edged away from a one-month trough against the yen. Markets have tended to see Clinton as the candidate of the status quo.
The president, who was born in Hawaii, released a long form birth certificate in 2011 to put the issue to rest. Only this month did Trump say publicly that he believed Obama was U.S.-born.
"He (Trump) has really started his political activity based on this racist lie that our first black president was not an American citizen. There was absolutely no evidence for it. But he persisted. He persisted year after year," Clinton said.
Trump repeated his false accusation that Clinton's failed 2008 presidential campaign against Obama had initiated the so-called "birther" issue.
"Nobody was pressing it, nobody was caring much about it ... I was the one that got him to produce the birth certificate and I think I did a good job," Trump said.
African-American voters overwhelmingly support Clinton, but Trump in recent weeks has said he believes his policy agenda would benefit them and said the policies of Obama and Clinton had failed to help black Americans.
He said Clinton's arguments were disingenuous.
"When you try to act holier than thou, it really doesn't work," Trump said.
Clinton wore a red pantsuit, and Trump wore a dark suit and a blue tie to the encounter that could shift the course of the tight race for the Nov. 8 election. She called him Donald. He called her Secretary Clinton for much of the debate before switching to her first name.
"She doesn't have the look, she doesn't have the stamina," he said.
Citing her own public record, Clinton retorted: "As soon as he travels to 112 countries and negotiates a peace deal, a ceasefire, a release of dissidents ... or even spends 11 hours testifying in front of a congressional committee, he can talk to me about stamina."
Each accused the other of distortions and falsehoods and urged viewers to check their campaign websites for the facts.
Clinton called the New York businessman's tax policies "Trumped-up trickle-down" economics and Trump accused the former secretary of state of being "all talk, no action."
"I have a feeling I'm going to be blamed for everything," Clinton, the first woman to win the presidential nomination of a major U.S. political party, said during one tough exchange.
Clinton knocked Trump for not releasing his income tax returns and said that decision raised questions about whether he was as rich and charitable as he has said. She noted that the few years of tax returns he had released showed that despite his wealth, he had paid no federal income tax.
"That makes me smart," Trump said.
"I have a tremendous income," he said at one point, adding that it was about time that someone running the country knew something about money.
Clinton criticized Trump for failing to pay some of the business people with whom his company had contracted. She said she had met a lot of people who had been cheated by her opponent.
Trump said such incidents of non-payment had taken place when the work was unsatisfactory.
Trump attacked Clinton for her trade policies and said she would approve a controversial trade deal with Asian countries despite opposing it as a candidate.
"You were totally in favor of it, then you heard what I was saying, how bad it is, and you said, 'Well, I can't win that debate,' but you know that if you did win, you would approve that," he said.
"Well Donald, I know you live in your own reality, but that is not the facts," she said.
Moderator Lester Holt struggled to rein in the candidates, with discussions about trade policy suddenly shifting to the fight against Islamic State as Trump accused Clinton of giving away information to the enemy by revealing on her website how she planned to defeat the group. Clinton said that unlike Trump, she at least had a plan for fighting Islamist militants.
Opinion polls have shown the two candidates in a very tight race, with the latest Reuters/Ipsos polling showing Clinton ahead by 4 percentage points, with 41 percent of likely voters.
A second Reuters/Ipsos poll released on Monday showed half of America's likely voters would rely on the debates to help them make their choice.
Two other presidential candidates - Libertarian Gary Johnson and Green Party candidate Jill Stein - were not invited to take part in the debate because neither had obtained at least 15 percent support in national polls, the threshold established to qualify.
Democrat Hillary Clinton accused Republican Donald Trump of racism, sexism and tax avoidance on Monday during a heated presidential debate that could reshape the 2016 campaign for the White House.
Trump and Clinton began their first face-off Monday in what many pundits presumed would be the most-watched debate in history ahead of the November 8 election.