Hillary Clinton was calm, cool, well-prepared and -- pundits say -- victorious in the first presidential debate against Donald Trump, but her strong showing may have little impact on polls showing them in a dead heat.

Both campaigns cried victory Tuesday after the sometimes nasty showdown in which the White House hopefuls sparred on the economy, crime, foreign policy and more personal issues -- like Clinton's email controversy and Trump's refusal to release his taxes.

"We had a great, great time last night," a beaming Clinton told reporters on a plane taking her to North Carolina for a renewed bout of campaigning.

"And I think people saw last night some very clear differences between us," the Democrat added.

The Republican Trump, who told AFP after the debate that it had gone very well for him, took to Twitter Tuesday to claim he had won every poll from the debate "except for the little watched @CNN poll."

That was a snap survey of 521 voters that said 62 percent found Clinton the winner against 27 percent for Trump.

But a CNBC online poll had it the other way round: 35 percent for Clinton compared to 65 percent for Trump.

"Clearly Clinton was by far the most informed and skilled debater," said Steffen Schmidt, a professor of political science at Iowa State University.

"Trump did not blow up and crash and burn and he acted more presidential than expected, except at the end when he was told that he supported the war in Iraq, when he began to harangue and bluster," Schmidt added.

Michael Heaney of the University of Michigan called Clinton's performance excellent.

"She stayed focused, and on message," he said.

Clinton also had more to say than Trump, who was on the defensive, not as well prepared, addressed issues only superficially and aimed a lot of personal attacks at Clinton.

In the debate's final stretch, Trump charged that "she doesn't have the stamina. To be president of this country, you need tremendous stamina."

In doing so, Trump laid himself open to a stringing retort from the former secretary of state who drew applause from the audience with her response.

"As soon as he travels to 112 countries and negotiates a peace deal, a cease fire, a release of dissidents, an opening of new opportunities in nations around the world, or even spends 11 hours testifying in front of a congressional committee, he can talk to me about stamina," she said.

For John Hudak of the Brookings Institution in Washington, "you saw not a perfect performance by Hillary Clinton, but as much as an imperfect performance as you can imagine by her opponent."

This first of three presidential debates was watched by more than 80 million people, US media reported citing an early Nielsen tally, in a very close race for the November 8 election.

Most American voters do not particularly like either candidate and many are still undecided.

Heaney said Monday's televised duel carried huge stakes for each candidate, and could have changed the direction of the race -- but ultimately did not.

"I don't think we'll see a big shift towards either candidate as a result of this debate," he told AFP.

Clinton's strong showing put the spring back in her step but she still faces a fierce battle in the six weeks to election day -- and her running mate Tim Kaine insisted they would not lower their guard.

"It will make some difference, but we have to make our case everyday," Kaine told MSNBC.

In a four way race that includes two minor party candidates, Clinton's poll lead has fallen from an average of seven points in early August to 1.6 points today. The race is also very close in battleground states where the election is likely to be decided.

Two more debates are scheduled for October 9 and 19.

For Sabato's Crystal Ball, a political newsletter from the University of Virginia, the consensus seems to be that Clinton won the first one -- but cautioned it would take until new nationwide and statewide polls come out at the end of the week to judge the impact, if any, on the race.

"We all know how many times Donald Trump has been counted out, only for him to endure," it said.

"We also know that Clinton and Trump are talking to two very different countries, and political analysts and reporters are generally in the country Clinton, not Trump, inhabits," it added.