The PLO executive committee vote was unanimous, said Palestinian cabinet minister Ibrahim Abu Najah. "That means no one will compete with him in the election for president."
The PLO chairmanship was one of three positions held by Arafat, and it was the most important.
He was also president of the Palestinian Authority (PA) and chief of the Fatah movement.
Abbas is to become head of the PLO, Palestinian parliament speaker Rawhi Fattuh has taken over as PA president until elections, and Faruq Qaddumi has been given leadership of Fatah.
The PLO has been Arafat's vehicle for leadership of the Palestinian movement since 1969.
Fattuh was to be sworn in on Thursday as the interim PA president, officials said.
General elections for president are to be held within 60 days.
Long-time Arafat confidant Qaddumi, who had opposed the 1993 Oslo accords with Israel that Abbas helped negotiate, cast his vote for Abbas by telephone from Paris.
Kaddumi had been considered a potential contender for the post.
Palestinian leaders signalled their determination to ensure a smooth transition by scheduling the swearing-in ceremony of Parliament Speaker Fattuh and electing a new PLO chairman just hours after Arafat's death.
Interim PA president Fattuh will
be sworn in on Thursday
Abbas has been acting as caretaker leader, along with Palestinian Prime Minister Ahmad Quraya.
"We can be certain the transition will be smooth and the Palestinian people deserve to have free and fair elections," Palestinian cabinet minister Saib Uraiqat said.
Abbas is likely to begin wielding effective power immediately, even while Fattuh holds the official reins.
Fattuh, 55, grew up in the Rafah refugee camp in Gaza.
He left Gaza for studies abroad in the 1960s, joined Fatah in 1968 and returned from exile in 1994, along with Arafat and other Palestinian officials.
In the first Palestinian general elections in 1996, he was elected to parliament on a Fatah slate. In 2003, he was appointed agriculture minister and a year later was chosen as speaker, replacing Quraya who became prime minister.