He won a decisive mandate to renew peace talks with Israel, rein in militants and try to end more than four years of Mideast bloodshed.
Israelis and Arabs are hoping the results will bring peace, reports CBS News Correspondent Howard Arenstein.
Addressing hundreds of cheering supporters two hours after polls closed, Abbas said major challenges lie ahead.
"There is a difficult mission ahead to build our state, to achieve security for our people, to provide a good life for our people, to give our prisoners freedom, our fugitives a life in dignity, to reach our goal of an independent state," he said.
Abbas said he was dedicating his victory to the memory of Yasser Arafat, to the Palestinian people, to those killing in fighting with Israel and to thousands of Palestinians in Israeli prisoners.
The victory of the staid Abbas, who has spoken out against violence and has the backing of the international community, ushered in a new era, after four decades of chaotic and corruption-riddled rule by Yasser Arafat who died Nov. 11.
Abbas, popularly known as Abu Mazen, has promised to reform the Palestinian Authority, overhaul the unwieldy Palestinian security services and quickly resume negotiations with Israel, stalled for four years.
His political objectives are the same as Arafat's: a Palestinian state in the West Bank, Gaza Strip and east Jerusalem, and a solution for Palestinian war refugees and their descendants.
After results of three exit polls were announced — giving Abbas between 66 percent and 69.5 percent of the vote — his supporters celebrated in the streets. In the West Bank city of Hebron, motorists honked horns and waved Abbas posters. In Ramallah, gunmen fired in the air in celebration.
Israeli Prime Minister Ariel Sharon expects to meet with Abbas soon, his aides said. Israeli officials said that in a gesture to Abbas, Israel plans to release some of the more than 7,000 Palestinian prisoners, provided Abbas stop militants from firing rockets at Israeli towns.
"I think this vote shows a change in the Palestinian street" moving away from support of violence, said Sharon aide Raanan Gissin.
Polls were open for 14 hours.
By 8 p.m. (1 p.m. EST), or one hour before the polls closed, 66 percent of 1.1 million registered voters had cast ballots, election officials said. The figure was expected to rise, since unregistered voters also participated in the election.
The election, the first presidential vote in nine years, proceeded largely without interruption. In one incident, gunmen fired in the air in an election office and in Jerusalem, voters complained of confusing arrangements.
Final results were to be announced Monday morning.
According to three exit polls, Abbas' main challenger, independent Mustafa Barghouti, won 20 percent, while the remaining five scored in the low single digits.
Analysts have said Abbas needs at least 60 percent support to resume negotiations with Israel. "He (Abbas) has a mandate from the voters," pollster Khalil Shekaki said of the exit polls.
However, Abbas faces a lengthy list of challenges. He must balance between Israel's demand to crack down on militants and his efforts to co-opt the gunmen. A major attack on Israel could undermine his credibility and sour peace hopes.
Major militant groups have indicated they are willing to halt attacks and give him a chance. However, the Lebanese guerrilla group Hezbollah, which funds some of the Palestinian militants, is trying to undercut Abbas, according to people close to the group.
Earlier this week, Hezbollah-funded gunmen with ties to Abbas' ruling Fatah movement killed an Israeli soldier in a West Bank ambush. On Sunday, Hezbollah carried out a deadly cross-border attack. An Israeli soldier, a French U.N. observer and a Hezbollah fighter were killed in the confrontation.
President Bush has said a resumption of peace talks must be accompanied by sweeping Palestinian reforms. U.S. Secretary of State Colin Powell, speaking on CNN, praised the vote as a "moment of opportunity for both sides."
The vote was witnessed by hundreds of foreign observers, including former U.S. President Jimmy Carter and former French Premier Michel Rocard.
Fearing a low turnout, the Central Election Commission kept the polls open two extra hours, until 9 p.m., citing logistical problems.
One election official said the move came amid heavy pressure from Fatah, which was concerned a low turnout could weaken Abbas.
In Jerusalem, Palestinians and international observers complained of confusion over registration lists, and Palestinians accused Israel of trying to intimidate them.
Backed by Fatah's formidable machine, Abbas was nearly assured of victory well before the vote when his toughest competition, jailed uprising leader Marwan Barghouti, pulled out and the largest opposition group, Hamas, declared a boycott.
Regardless, many saw the vote, which Arafat had repeatedly delayed, as a hopeful sign the Palestinians were building a democratic foundation. Palestinian leaders have already scheduled a parliamentary election for July 17.
Many Palestinian refugees outside the West Bank and Gaza complained they could not vote for the person who would represent them in peace talks. The refugee issue has been a key sticking point in past peace talks, with Palestinians demanding they be allowed to return to the homes inside Israel that they fled or were driven from during the 1948 Middle East War.
Israel says it will not allow them to return, frightened their numbers would overwhelm the country and damage its Jewish character.
"How can they ignore our right to pick our leader?" asked Haitham Abul-Saad, 30, in the Baqaa camp, 17 miles northwest of the Jordanian capital Amman.
But eligible voters cast their ballots with great hope. "The election is our weapon to change our lives," said Souad Radwan, a 46-year-old teacher in Gaza's Jebaliya refugee camp, whose house was demolished during a recent Israeli raid into the camp. "We are sick of the occupation and this troubled life."