Palestine has had non-member observer state status since 2012, which allows some rights short of a full member.
Membership can only be decided upon by the UN Security Council.
The US recently vetoed a bid for full membership, but Friday's vote can be seen as a gesture of support for the Palestinians.
Palestinian President Mahmoud Abbas welcomed the resolution, saying it supported Palestinian efforts for another vote on the issue by the Security Council."Palestine will continue its endeavour to obtain full membership in the UN," he said in a statement.
Israel's ambassador to the UN, Gilad Erdan, said the body had welcomed a "terror state" into its ranks.
Addressing the assembly, he shredded a copy of the UN Charter - accusing members of having metaphorically done just that by passing the resolution.
It comes amid reports that several European countries plan to recognise a Palestinian state.
EU foreign policy chief Josep Borrell told Spanish broadcast RTVE on Thursday that Spain would do so on 21 May. He has previously said Ireland, Slovenia and Malta would also take the step, without confirming the date.
Friday's UN resolution confers additional rights on Palestine at the world body, allowing it to take part fully in debates, propose agenda items and have its representatives elected to committees.
It will still not, however, have the right to cast a vote - something the General Assembly does not have the power to grant and would have to be backed by the Security Council.
The issue of Palestinian statehood has vexed the international community for decades.
In 1988, the Palestine Liberation Organisation (PLO), the main representative of the Palestinians, first declared the establishment of the State of Palestine.According to the Reuters news agency, Palestinian statehood has been recognised by 139 out of 193 UN member states - although this is largely seen as symbolic.
In practice, the Palestinians have limited self-government through the Palestinian Authority (PA) in parts of the Israeli-occupied West Bank. The PA lost control of the Gaza Strip to Hamas in 2007. The UN considers both territories as occupied by Israel and comprising a single political entity.
Israel does not recognise Palestinian statehood and the current Israeli government opposes the creation of a Palestinian state in the West Bank and Gaza. It argues such a state would be a threat to Israel's existence.
The US endorses the creation of an independent Palestinian state alongside Israel - the so-called two state solution to the Israel-Palestinian conflict - but says such a state should only come through direct negotiations between the two sides.
Last month, the US used its veto as one of the five permanent members of the Security Council to block a widely backed Algerian resolution seeking Palestine's admittance as a state, calling it "premature".Security Council resolutions are legally binding, whereas General Assembly resolutions are not.
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