Abbas won 66.3 percent of the votes cast while Barghuti, an independent candidate, registered 19.7 percent of the votes, said the study by the Palestinian Centre for Policy and Survey Research released shortly after the official close of voting at 1900 GMT.
Former prime minister Abbas, who was running for the dominant Fatah faction, had been widely expected to be elected as the successor to the late Yasser Arafat.
A total of seven candidates took part in the election although none of the others registered more than four percent of the vote, the exit poll said.
Tayssir Khaled, standing for the leftist Democratic Front for the Liberation of Palestine, registered 3.8 percent, while former communist People's Party candidate Bassam Salhi had 2.7 percent. The independents Abdelhalim Al-Ashqar, Sayyed Barakah and Abdelkarim Shubeir won 2.4, 1.1 and 0.7 percent, respectively.
A formal announcement of the result was not expected to be made until Monday morning.
Abbas's bid to usher in a new era of diplomacy will be vulnerable to freedom fighters, who boycotted the vote and fired two rockets into Israel during polling in a show of force against his calls for a cease-fire.
Abbas supporters celebrated in the streets of Ramallah, honking car horns, waving flags and holding his portrait aloft.
Hamas, which boycotted the Palestinian presidential election, remained tight-lipped over the exit poll.
"We will wait for the final result to give a reaction. We don't want to speak on this matter before the final result," spokesman Mushir al-Masri told AFP in Hamas' stronghold in the volatile Gaza Strip.
Hamas had said it would co-operate with the victor, whoever won the election, despite criticising the poll.
Its smaller rival, Islamic Jihad, which similarly snubbed the vote, stressed it would deal with Abbas as it worked with the late Yasser Arafat and called on the Palestinian uprising against Israel to continue.
"We hope that he will start to prepare Palestinian affairs tomorrow, re-start dialogue for a Palestinian national leadership... and not to take this victory to fight or bow to Israeli and American pressure," spokesman Khaled el-Batsch told AFP.
Palestinian election officials decided in mid-afternoon to extend polling in the West Bank, Gaza Strip and East Jerusalem by two hours to 9 pm because, they said, some voters were being held up by Israeli army checkpoints.
Monitors said turnout was depressed by strict Israeli limits on polling stations in East Jerusalem, which Israel annexed after capturing it along with the West Bank and Gaza in the 1967 Middle East war but which Palestinians want for their capital.
But they said Israel appeared to have largely kept its promise to ease the passage of Palestinians through checkpoints.
"Anecdotal evidence coming in is that restrictions have been quite effectively lifted," said Les Campbell, a spokesman for foreign observers led by former US President Jimmy Carter.
"(Overall) it has been a very good day. The moment is historic," European Union foreign policy chief Javier Solana said while observing voting.
The voting extension followed reports of low turnout in cities such as East Jerusalem, Nablus and Ramallah - a cause for concern in Abbas's camp that he might fall short of a broad popular mandate needed to swing freedom fighters behind a peace push.
It remained unclear whether lagging turnout arose from boycott pressures by Islamists who score about 25 percent in opinion polls, travel restrictions or apathy since a victory for Abbas seemed to many Palestinians a foregone conclusion.
Opinion polls had suggested Abbas would win between 50 and 65 percent of the vote.
Israel, who has sized up Abbas as a man they can do business with but criticised his intention to co-opt rather than confront freedom fighters, reasserted that progress towards peace depended on a halt to "terrorism and violence".
After decades as a backroom technocrat in Arafat's circle, Abbas waged a populist campaign pledging to uphold his old guerrilla boss's quest for total Israeli withdrawal from occupied lands but parted from him in urging an end to violence.
Abbas caused disquiet in Israel with campaign vows to insist on Palestinian statehood in all of the West Bank, Gaza and East Jerusalem, as well as a "right of return" for millions of Palestinian refugees to lands now inside the Jewish state.
Israel plans to withdraw 8,000 Jewish settlers from tiny Gaza this year. But it rules out ceding East Jerusalem or taking back refugees and has won US assurances that it should never have to give up much larger settlements in the West Bank.
On the Israeli political front, a new government was to be sworn in on Monday after Prime Minister Ariel Sharon's coalition pacts with veteran peacemaker Shimon Peres's Labour Party and a religious faction were submitted to parliament on Sunday.
With new partners, Sharon will have a parliamentary majority for the first time in six months to press ahead with the removal of all 21 Gaza settlements and four of 120 in the West Bank.