Israel criticised the move and voted against it, along with the United States and six other countries.
The Israeli and Palestinian leaders will speak before the UN this week.
"The sense of pride among the Palestinian people was overwhelming the day the world voted in favour of this landmark initiative," Palestinian Authority President Mahmoud Abbas wrote in the Huffington Post on Tuesday.
"I am certain that the day our flag rises among the flags of the community of nations will also be a most emotional and proud day."
He also called for a multilateral solution to the Israeli-Palestinian conflict, adding: "The UN must give my people more than hope."
The motion was passed by 119 votes, but 45 countries abstained from the vote.
At the time, Israel`s permanent representative to the UN, Ron Prosor, said the move was a "blatant attempt to hijack the UN". He insisted that the only way Palestinians could achieve statehood was through direct negotiations.
In 2012, the UN General Assembly voted to upgrade the status of the Palestinians to that of a "non-member observer state" - the same position that the Vatican holds.
It followed a failed bid to join the international body as a full member state in 2011 because of a lack of support in the UN Security Council.
Mr Abbas will address the General Assembly in New York on Wednesday, with Israeli Prime Minister Benjamin Netanyahu, speaking a day later.
He is expected to call on Palestinians to end "incitement to violence".
In recent weeks, there have been clashes between Palestinian protesters and Israeli police at a disputed holy site in Jerusalem, known to Muslims as the Haram al-Sharif and to Jews as the Temple Mount.
Early on Wednesday, Israel carried out a series of air strikes on Gaza, hours after the Israeli Iron Dome anti-missile system intercepted a rocket from the enclave.
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