Polling closed at 1900 GMT, two hours later than scheduled after voting difficulties forced an extension.
One exit poll, by the Palestinian Centre for Policy and Survey Research, gave Mr Abbas 66% of the vote and his closest challenger, Mustafa Barghouti, 19.7%.
A high margin of victory, as well as good voter turnout, are considered crucial if Mr Abbas is to command enough of a mandate to push through his agenda of peace talks and an end to the armed Palestinian uprising.
Although a final result is not expected until tomorrow, supporters of Mr Abbas are already celebrating.
This is the first presidential election for Palestinians since 1996, and about 1.1 million people registered to vote.
There were problems with registration, heavy turnout and the turning away of hundreds of voters from a big Israeli-run polling station in East Jerusalem.
East Jerusalem has special voting arrangements, with Palestinians allowed to vote in Israeli post offices.
Voters complained Israeli officials were not allowing them to vote even though the Palestinian central election commission had properly registered them.
Mr Abbas himself cast his vote at the Ramallah compound where the late leader of the Palestinians Yasser Arafat is buried. He said the vote was taking place in a “marvellous” fashion.
“It is an illustration of how the Palestinian people aspire to democracy,” he said.
The death of Mr Arafat in November last year has raised hopes for new peace talks between the Palestinians and Israelis.
The Israeli Prime Minister, Ariel Sharon, had refused to do business with Mr Arafat.
Mr Abbas, also known as Abu Mazen, is the nominated candidate and a founding member of the main Palestinian political faction, Fatah.
He is widely regarded as a moderate and is considered to have been the main architect of the Oslo Peace Accord which came close to resolving the conflict in the early 1990s.
He served as prime minister to Yasser Arafat in 2003 but resigned again after four months of a power struggle in which Arafat refused to hand over crucial powers to him.
He became chairman of the Palestinian Liberation Organisation after Arafat’s death and has been the front-runner to succeed him ever since.
Courtesy BBC News
Mahmoud Abbas was formally sworn in as president on 15 January.
Despite Mr Abbas’ support for a return to peaceful resistance, attacks by militant groups continued after his election.
However, Mr Abbas deployed Palestinian police in northern Gaza and by February had persuaded Hamas and Islamic Jihad to begin a temporary, unofficial cessation of violence.
Mr Abbas and the Israeli Prime Minister Ariel Sharon went on to announce a mutual ceasefire at a summit in Egypt, although the militant groups stopped short of making it official.
In August, Israel pulled out of the Gaza Strip, with emotional scenes as Israeli troops removed some settlers by force.
Mr Sharon’s policies caused bitter resentment within his party, Likud, and in November 2005 he resigned to form a new centrist party of his own.
After suffering a stroke on 4 January 2006, Sharon remained in a permanent vegetative state until his death in January 2014.