Palestinian President Mahmoud Abbas told the assembly the vote was the "last chance to save the two-state solution" with Israel.
Israel's envoy to the UN said the bid pushed peace process "backwards", while the US said the move was "unfortunate".
The assembly voted 138-9 in favour, with 41 nations abstaining.
Hundreds of Palestinians celebrated on the streets of Ramallah, in the West Bank, after the result was announced.
'Birth certificate'
"Sixty-five years ago on this day, the United Nations General Assembly adopted resolution 181, which partitioned the land of historic Palestine into two states and became the birth certificate for Israel," Mr Abbas said shortly before the vote in New York.
"The General Assembly is called upon today to issue a birth certificate of the reality of the State of Palestine," he said.
The Israeli ambassador to the UN, Ron Prosor, said "the only way to reach peace is through agreements" between the parties, not at the UN.
"No decision by the UN can break the 4,000-year-old bond between the people of Israel and the land of Israel," he said.
Opponents of the bid say a Palestinian state should emerge only out of bilateral negotiations, as set out in the 1993 Oslo peace accords under which the Palestinian Authority was established.
Speaking after the vote, the US ambassador to the UN, Susan Rice, urged the Palestinians and Israel to resume direct peace talks and warned against unilateral actions.
US Secretary of State Hillary Clinton called the vote "unfortunate and counter-productive", saying it put more obstacles on the path to peace.
"By going to the UN, the Palestinians have violated the agreements with Israel and Israel will act accordingly," said the office of Israeli Prime Minister Benjamin Netanyahu on Twitter.
UN Secretary General Ban Ki-moon also called for more talks, saying the resolution underscored the need to resume meaningful peace negotiations.
The UK abstained from the vote, as did Germany. The Czech Republic, the Marshall Islands and Panama were among the nations voting with the US and Israel.
In the West Bank, crowds celebrated the vote by waving flags and chanting "God is great!"
"For the first time, there will be a state called Palestine, with the recognition of the entire world," Amir Hamdan was quoted as saying by the Associated Press news agency.
"Today the world will hear our voice," he added.
Symbolic milestone
The Palestinians are seeking UN recognition of a Palestinian state in the West Bank, Gaza and East Jerusalem, the lands Israel captured in 1967.
While the move is seen as a symbolic milestone in Palestinian ambitions for statehood, the "Yes" vote will also have a practical diplomatic effect, says the BBC's Barbara Plett, at the UN.
It would allow the Palestinians to participate in debates at the UN and improve their chances of joining UN agencies and bodies like the International Criminal Court.
Last year, Mr Abbas asked the UN Security Council to admit the Palestinians as a member state, but that was opposed by the US.
Mr Abbas was much criticised by many Palestinians for remaining on the sidelines of the conflict earlier this month in Gaza and efforts to achieve a ceasefire with Israel.
His Fatah movement, based in the West Bank, is deeply split from the militant Hamas movement which governs Gaza.
Gaza's Prime Minister Ismael Haniyeh said in a statement sent to the BBC that Hamas' "support for the UN bid is based on the 'rule of non-recognition of the occupier'... and the right of Palestinians to return to their homeland".
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