Israeli leaders welcomed Abbas' victory, but said they will watch closely how hard he tries to subdue militants. Palestinians celebrated the results.
"Today is the beginning of a new future," said Sami Radwan, 55, a restaurant owner in Gaza City. "Abu Mazen [Abbas] is the right choice. He is the one who can bring us peace, good business and security."
Palestinians say Abbas' first priority should be resuming talks with Israel, reports CBS News Correspondent Howard Arenstein.
"I think he should extend an invitation for an immediate resumption of the peace negotiations," said Palestinian Ghassan Khatib, adding that Israel has to do its part to improve living conditions in the West Bank and Gaza Strip, or else militants will continue their attacks.
"As long as they see the continuous expansion of settlements, the continuous expansion of this occupation, they will continue resisting this occupation," Khatib said.
In other developments:
Israeli security forces in jerusalem were on high alert, reports Arenstein, with troops and police sweeping the area in search of a suicide bomber.
Abbas could easily lose his political capital over a major bombing or shooting attack, and while most militant groups signaled they are willing to give him a chance, not all have signed on to a truce with Israel.
Still, Abbas' victory in Sunday's vote held out the promise of a new era after four decades of chaotic and corruption-riddled rule by Yasser Arafat, who died Nov. 11. Abbas, who has spoken out against violence and has the support of the international community, promises to reform the government and the unwieldy security services.
President Bush said Monday he would welcome newly Abbas to the White House.
Abbas won 62.3 percent of the vote, according to final results announced by Hanna Nasser, head of the Central Election Commission.
His main challenger, independent Mustafa Barghouti, won 19.8 percent, while the remaining five candidates scored in the lower single digits. About 3.8 percent of the ballots were deemed invalid, and 3.2 percent were blank, Nasser told a news conference.
Nasser declined to give a turnout figure, citing confusion over the use of outdated residency records.
Questions about voter participation are a possible point of contention between Abbas' Fatah movement, which was pushing for a high turnout, and the Islamic militant group Hamas, which had called for a boycott.
In his acceptance speech, Abbas said he faces a difficult mission, but he reiterated that he would not go after militants. Instead, he said, he wants to "give our fugitives a life of dignity," referring to those wanted by Israel.
"I present this victory to the soul of Yasser Arafat and present it to our people and to our martyrs," Abbas added.
After exit polls predicted a sweeping Abbas victory, cheering supporters took to the streets of the West Bank and Gaza late Sunday. Gunmen fired in the air, motorists honked horns and members of Abbas' ruling Fatah movement, wearing checkered black-and-white headbands, danced in the streets.
Hamas, the largest opposition group, announced Monday it will work with Abbas, despite misgivings about what it said were voting irregularities, including a decision to keep polls open two hours longer than planned. Hamas had called for a boycott of the election, but did not try to disrupt the vote.
"This opens up an opportunity, in my opinion, for renewed peace negotiations, as well as justice, long overdue for the Palestinian people," said former president Jimmy Carter, who led the international monitoring team.
Carter hopes the election leads to two sovereign nations, "hopefully ultimately becoming friends living completely at peace. That's my dream and my hope and my prayers."
Labor leader Peres praised Abbas as a wise leader, and expressed hope that peace talks could resume with new Israeli and Palestinian governments.
Peres congratulated Abbas in a telephone call and told the Palestinian leader he would do everything he could to help, said an official close to Peres.
Sharon plans to meet with Abbas soon, the Israeli leader's aides said.
Most Palestinian militant groups have indicated they are willing to consider a halt to attacks against Israel.
However, Lebanese Hezbollah guerrillas, who fund some Palestinian militants, are trying to sabotage a possible truce, according to people close to the group.