Speaking yesterday to supporters in the West Bank town of Ramallah, Abbas dedicated his victory to his predecessor Arafat, who died in November.
"We offer this victory to the soul of the brother martyr, Yasser Arafat, and we must pray on his soul," Abbas said.
Exit polls show Abbas, the candidate of Arafat's Fatah movement, winning more than 65 percent of ballots. His nearest rival was pro-democracy activist Mustafa Barghouthi, who had about 20 percent. Abbas is to be inaugurated on 12 January.
The election of Abbas, a former prime minister, has rekindled hopes that peace talks can resume after four years of bloodshed. He has repeatedly called for an end to the Palestinian uprising against Israel.
His victory, meanwhile, coincides with a key development in Israel, where a new cabinet is set to take office today. Prime Minister Ariel Sharon's government will seek to pull Jewish settlements out of Gaza and parts of the West Bank. The government will include the Labor Party of Shimon Peres.
Peres, set to be deputy prime minister, said today the election of Abbas marks "the beginning of a new process."
In Washington, U.S. President George W. Bush called the vote "historic" and said he was "heartened" by the strong voter turnout, reported at about 65 percent.
Abbas, also known as Abu Mazen, hailed the election as a sign of democratic progress. But he warned of challenges ahead and repeated he would not use force to go after militants.
"There are difficult tasks ahead: How are we going to build a state of security and safety? How do we find an honorable life for our citizens? How shall we solve the issue of prisoners [in Israeli jails]? How to solve the issue of our fugitives? How do we solve the problem of our Palestinian state, with Jerusalem as its capital?" Abbas said.
Election officials extended voting by two hours in the West Bank, Gaza Strip, and East Jerusalem. They said some voters were held up at Israeli checkpoints in areas occupied since the 1967 Middle East war.
Former U.S. President Jimmy Carter, working as an election monitor, said he intervened directly with Sharon's advisor, Dov Weisglass, about the problems.
"Last night, I called about one of the checkpoints near Ramallah which was holding people up. And it was immediately opened up. So I have his private phone number. And during the day, if anything should arise that is a problem, I have the authority to call the [Israeli] prime minister's office directly," Carter said.
Aides to Sharon say the prime minister plans to meet with Abbas soon. Abbas has called Sharon a potential "partner" for peace.
Most militant groups have suggested they are willing to halt attacks against Israel to give Abbas a chance to start peace talks.
However, the Lebanese-based Hizballah guerrillas, who fund some Palestinian militants, are reportedly seeking to hamper attempts by Abbas to attain a ceasefire.
Yesterday, Hizballah launched a cross-border attack. The exchange resulted in the deaths of an Israeli soldier, a French United Nations observer and a Hizballah fighter.