Abbas demanded the United Nations give a "birth certificate" to a Palestinian state and was rewarded with the backing of 139 countries.
Only nine members heeded Israeli warnings that the move could lead to more violence and voted against. Forty-one abstained.
The vote lifts the status of the Palestinian Authority from an observer entity to a "non-member observer state" with the same status as the Vatican.
Even though it is not a full member it can now join UN agencies and potentially join the International Criminal Court.
The Palestinian leadership says it wants to use the "historic" vote as a launchpad for talks with Israel which have been frozen for more than two years.
Abbas, who embraced his foreign minister after the vote and was given repeate standing ovations, said the vote was "the last chance to save the two-state solution."
In a 22-minute speech laced with references to Israel's assault this month against rockets fired from Gaza, Abbas said Palestinians would accept "no less than the independence of the state of Palestine with East Jerusalem as its capital."
He added: "We must repeat here once again our warning: the window of opportunity is narrowing and time is quickly running out. The rope of patience is shortening and hope is withering."
Abbas said UN members had to "issue a birth certificate of the reality of the state of Palestine."
US ambassador Susan Rice condemned the vote as "an obstacle to peace" because it would not lead to a return to direct talks between the Israelis and Palestinians.
"Today's grand pronouncements will soon fade and the Palestinian people will wake up tomorrow and find that little has changed," she told the assembly, in a grimly delivered statement.
"This resolution does not establish that Palestine is a state."
The United States blocked a Palestinian application for full membership of the United Nations that Abbas made in September 2011.
"The UN resolution will not confer statehood on the Palestinian Authority," Israeli ambassador to the United Nations Ron Prosor said.
He added that making Palestine a non-member observer state at the UN "will place further obstacles and preconditions to negotiations and peace." He warned that it could lead to increased violence.
Abbas was warned earlier by UN leader Ban Ki-moon that the Middle East peace process is on "life support" and Israeli Prime Minister Benjamin Netanyahu also said the UN General Assembly vote would not create a Palestinian state.
Ban urged both sides to return to talks which currently look a distant prospect, diplomats said.
The Palestinian leader did not make any reference to the possibility of joining the International Criminal Court -- a major worry for Israel.
But Abbas said the Palestinian Authority would consult with other countries about new steps after its diplomatic status is bolstered.
"We will act responsibly and positively in our next steps, and we will to work to strengthen cooperation with the countries and peoples of the world for the sake of a just peace," he said.
Talks between the two have been suspended since September 2010, with the Palestinians blaming Israeli settlements in the occupied territories.
The vote comes 10 days after a ceasefire ended a brief but bloody conflict between Israel and Hamas, the Palestinian militant group that holds sway in the Gaza Strip and is a rival of Abbas and his West Bank-based Fatah faction.
The landmark General Assembly meeting was held on the 65th anniversary of a UN resolution on the division of the Palestinian territories into a two-state solution that Ban said "remains tragically unfulfilled."
The Palestinians say 132 countries now recognize their state bilaterally and said the result was a boost. Several countries which do not recognize the Palestinian state, such as France, voted for the resolution.
But several European countries, including some backing the bid, believe the Palestinians should have waited until after US President Barack Obama installed his new administration and Israel held elections, diplomats said.
Success gives the Palestinians access to UN agencies and treaties but there are divided opinions over whether they will be able to automatically join the ICC.
Palestinian envoys have said Abbas will not rush to join the court but could use it if Israel does not change its policies on settlements and other matters.
The Palestinian Authority and UN agencies that accept Palestinian participation could lose hundreds of millions of dollars in financing because of the vote.
US law prohibits funding for any international body recognizing a Palestinian state.
Washington has warned Abbas he risks losing around $200 million in aid, which is blocked in the US Congress.
Israel is considering freezing the transfer of tax and customs funds it collects for the Palestinians, while one Israeli foreign ministry policy paper even suggested "toppling" the Palestinian Authority.