A jubilant Mr Abbas proclaimed victory soon after exit polls showed he had gained around two-thirds of the vote, with a turnout of around 68 percent.
A final result will be officially announced on Monday morning, but the outcome of only the second-ever presidential election is clear.
“We dedicate this victory to the memory of our martyred leader Yasser Arafat, as well as all the other martyrs, those who have been wounded as well the 11,000 prisoners behind bars” in Israeli jails, Mr Abbas told hundreds of his supporters in the Palestinian Authority’s political capital Ramallah.
“I will work to put an end to the suffering of the Palestinian people for they are a people who deserve our esteem, our respect and our loyalty.”
He called on militant Palestinian groups, who boycotted the elections, to end their armed uprising against Israeli occupation.
His nearest rival, independent Mustafa Barghouti, scored around 20 percent of the vote.
As Mr Abbas toasted his victory inside a meeting hall, gunmen from his Fatah party were seen driving around the streets of Ramallah firing in the air to celebrate his triumph.
Voting was extended for two hours after the central elections commission claimed that Israeli restrictions in the occupied territories had hampered access to polling booths.
However monitors said that by and large there had been no major problems and there were no reports of any violence.
While a Hamas representative said Mr Abbas will not succeed as Israel will not give him a chance, Hamas said it would work with him.
An Israeli foreign ministry spokesman said Israel regards Mr Abbas as a man it can negotiate with.
Israeli Labour leader Shimon Peres said his election will mark the beginning of a new process.
“Mr Abbas will be an uncompromising partner, but he is a wise man, experienced and moderate,” he said.
“He was chosen by the vast majority of Palestinians and we should give him the chance to succeed,” said Mr Peres.
He said he hopes the new leadership will be a “reflection of a change in the Palestinian path”.
US President George W Bush hailed the vote as an historic day for the Palestinian people.
“Palestinians throughout the West Bank and Gaza took a key step toward building a democratic future by choosing a new president in elections that observers describe as largely free and fair,” he said in a statement.
Throughout campaigning, Mr Abbas called for the creation of an independent Palestinian state with east Jerusalem as its capital, and the right of return for Palestinian refugees, in line with the Palestinian movement.
However he also made some controversial comments criticising the armed Palestinian uprising, and also indicated his wish for reform and end the security chaos in the region.
An Abbas presidency is likely to lead to a resumption of top-level talks with Israel, frozen since he walked out after a short stint as Mr Arafat’s first premier in September 2003.