Abbas, a long-time deputy of Arafat, reiterated the goal of creating a state independent of Israel, while Israeli officials urged him to crack down on militants to allow the peace process to proceed.
In the election considered by many Palestinians and international officials as a milestone in bringing democracy to the Arab world, Abbas -- the candidate of the dominant Palestinian political party Fatah -- won 65 percent of ballots cast, election officials said.
The next leading candidate, physician and human rights activist Mustafa Barghouti, who was running as an independent, received 21 percent of the vote, according to The Washington Post.
Sunday's vote, the first in nine years, was to elect the president of the Palestinian Authority who governs much of the West Bank and Gaza Strip. Abbas, or Abu Mazen as he is more commonly known, succeeds Arafat who died two months ago after leading the Palestinians for nearly 40 years.
Abbas had declared victory late Sunday after exit polls gave him a commanding lead.
"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," Abbas said, reported CNN.
Israeli Deputy Prime Minister Ehud Olmert said Israel is ready to make "all the necessary adjustments" to work with Abbas.
"The main challenge is still ahead for him," Olmert said. "Will he fight against the terrorists? Will he try to stop this bloody, violent war against the state of Israel? This is the main question. This is what interests us."
President Bush welcomed the election, calling it "a historic day for the Palestinian people and for the people of the Middle East.
"America and all free nations strongly support the efforts of the Palestinian people to create lasting democratic institutions," the president said in a statement Sunday. "These efforts -- including today's presidential elections and the parliamentary elections that will follow in several months -- are essential for the establishment of a sovereign, independent, viable, democratic, and peaceful Palestinian state that can live alongside a safe and secure Israel."
International and Palestinian election monitors reported widespread confusion at polling stations in East Jerusalem, where they said Israeli officials severely limited the number of Palestinians allowed to cast votes, The Washington Post reported.
The monitors said, however, that Israelis largely adhered to commitments to ease travel restrictions at checkpoints throughout the West Bank, and no significant problems were reported in the Gaza Strip.
Overall, the election was marked by far fewer problems than many Palestinian organizations had predicted, although voter turnout was so low at mid-afternoon that the Palestinian Election Commission extended voting by two hours.
Abbas is expected to be sworn in as head of the Palestinian Authority at a session of parliament on Wednesday.
Rawhi Fattouh, who has been acting as head of the Palestinian Authority since Arafat's death on Nov. 11, will resume his role as Palestinian Legislative Council speaker at the same parliamentary session.
-- Compiled from wire reports and other media sources