According to the initial results that were released on Monday, Abbas won 66.7% of the West Bank's votes and 65% of the Gaza Strip's.
Palestinian National Initiative candidate Mustafa al-Barghuthi won 21% of the West Bank votes and 18% of the Gaza Strip's. Abbas declared himself the winner early on Monday.
The Palestine Liberation Organisation chairman immediately dedicated his victory to late leader Yasir Arafat.
In a first reaction to the results, independent candidate Sayid Husayn Baraka has called for repeating the election, adding that forgery was widespread.
Mustafa al-Barghuthi won 21%
of the West Bank votes
"Forgery took place when the voting time was expanded for two hours," he said.
The elections were not based on equal chances and the Israeli forces did not ease the process as it had previously vowed to do, blocking him from entering the West Bank until six days before the polls, Baraka said.
"Unfortunately, these elections are not honest for many reasons," he said.
"In addition, an unlimited number of citizens said yesterday [on Sunday] they voted more than once. Therefore, I don't accept the results of these elections."
Nevertheless, with a final official result yet to be announced, Abbas said he would "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 also vowed not to forget "all the other martyrs, those who have been wounded as well the 11,000 prisoners behind in Israeli jails".
Early exit polls indicated 65% of the electorate turned out to vote, despite a boycott by some resistance groups opposed to Abbas' call for an end to armed resistance.
Abbas campaign manager Muhammad Shataya said the result meant "that Abu Mazin [Abbas] has the mandate to implement his programme".
But not everyone was happy with the result, and one resistance group fired rockets into Israel during polling.
"It has been a very good day. The moment is historic"
EU foreign policy chief
Shaikh Hasan Yusuf, a spokesman for the Hamas resistance group in the occupied West Bank, said he respected the people's choice.
"But for Abu Mazin to get 70% of the votes of the eligible voters means he got the support of some 35% of the Palestinians. This is very weak," he added.
Five other presidential candidates, ranging from a former resistance fighter to an academic under US house arrest, trailed far behind.
The relatively high turnout of 65% appeared to quell concerns that low participation due to a boycott call could dent Abbas' mandate for peace talks and internal reform.
Palestinian election officials extended polling in the West Bank, Gaza Strip and East Jerusalem by two hours because some voters were being delayed at Israeli checkpoints in the territories occupied since the 1967 Middle East war.
But monitors said Israel appeared to have largely kept its promise to ease the passage of Palestinians through checkpoints.
"It has been a very good day. The moment is historic," European Union foreign policy chief Javier Solana said while observing the voting.