The poll said the law professor, 61, had secured 76% of votes. No official results have yet been published.
He was up against media mogul Nabil Karoui, 56, who campaigned from prison after being arrested on charges of money laundering and tax fraud.
He denies the charges and officials have warned he may appeal, arguing he has not been allowed a fair campaign.
After the exit polls were released, Mr Saied appeared in front of jubilant supporters in Tunis and thanked “young people for turning a new page” in Tunisia’s history.
“We will try to build a new Tunisia. Young people led this campaign, and I am responsible for them,” he said, flanked by his family.
Mr Karoui, who was freed by a court order just four days ago, came second in the first round of voting, with 15.6% of the vote to Mr Saied’s 18.4%.
Supporters take to the streets
Rana Jawad, BBC North Africa correspondent
With Kais Saied apparently leading by a large majority, his supporters took to the streets in celebration. Official results will be revealed on Monday.
Kais Saeid and his rival Nabil Karoui held a debate on Friday night that was watched by more than six million Tunisians.
Mr Saied provided little policy detail, merely speaking of empowering the people and being a man of the law, leaving his critics and undecided voters with the impression that he has no plan.
Mr Karoui, meanwhile, said he would alleviate poverty and bring in high-tech jobs from Google, Netflix and Amazon – a promise that was the subject of much satire on social media.
The debate, however, did appear to inject more energy into the second round of voting.
Who is Kais Saied?
Nicknamed “the robot” for his stern manner, Mr Saied has run a shrewd campaign, with almost no advertising, on a message of integrity and anti-corruption targeted at young Tunisian voters.
Mr Saied was on the committee of experts that helped parliament draft Tunisia’s post-Arab Spring constitution, adopted in 2014. He occasionally appeared on television as a political commentator.
In the week before the polls, he announced that he would not campaign while his rival was in prison.
He has promised electoral reforms, including changes to local elections for regional representatives.
Critics have attacked his conservative social views, however.
In an interview with a local newspaper, he accused foreign powers of encouraging homosexuality in the country.
He is in favour of restoring the death penalty, suspended since 1994 in Tunisia.
If he wins the presidency, his wife will not be known as the country’s First Lady, he says. He opposes equal inheritance for men and women.
What is the significance of the poll?
The election was brought forward after the death in July of Tunisia’s first democratically elected president, Beji Caid Essebsi.
He took office in 2014, three years after a popular uprising that led to the overthrow of long-serving ruler Zine al-Abidine Ben Ali and sparked the Arab Spring.
Correspondents says that while the Arab Spring gave Tunisia democracy eight years ago, many believe it brought little else.
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