The Palestinian Authority had initially agreed to defer a vote on the UN-sanctioned report, but later backtracked under heavy criticism.
The United States and Israel were among those countries which voted against the resolution.
Mike Hanna, Al Jazeera's correspondent in Geneva, said the vote was a "very strong victory" for the supporters of the resolution, but that the large number of abstentions was also "very significant".
In Gaza, Hamas thanked the nations that voted to endorse the report.
"We hope that vote will lead to a trial of the occupation leaders," Taher al-Nunu, a Hamas spokesman, said.
Mousa Abu Marzook, the deputy chairman of the Hamas political bureau in Damascus, Syria, told Al Jazeera: "We thank our people, all those who support to submit again this report to the human rights committee and all the countries who voted for the report.
"I think if the Palestinian Authority didn't withdraw this report it will be more efficient and the result will be stronger than the resolution.
"We will co-operate with this report and we will establish a new committee to investigate.
"Right now, there is no talking with Fatah, but during the dialogue between Fatah and Hamas in Egypt, within a few weeks, we are going to talk about reconciliation and, of course, this kind of subject we are going to talk about."
In addition to endorsing the report, the resolution "strongly condemns all policies and measures taken by Israel, the occupying power, including those limiting access of Palestinians to their properties and holy sites".
For: Argentina, Brazil, China, Russia, Bahrain, Bangladesh, Bolivia, Chile, Cuba, Djbouti, Egypt, Ghana, India, Indonesia, Jordan, Mauritius, Nicaragua, Nigeria, Pakistan, Philippines, Qatar, Saudi Arabia, Senegal, South Africa and Zambia
Against: US, Hungary, Italy, Netherlands, Slovakia and Ukraine
Abstentions: Belgium, Bosnia, Burkina-Faso, Cameroon, Gabon, Japan, Mexico, Norway, South Korea, Slovenia and Uruguay
No vote: UK, France, Madagascar, Kyrgyzstan and Angola
It also calls on Israel to stop digging and excavation work around the Al-Aqsa mosque as well as other Islamic and Christian religious sites.
Israel rejected the charges saying the resolution – drafted by the Palestinians with Egypt, Nigeria, Pakistan and Tunisia, on behalf of non-aligned, African, Islamic and Arab nations – threatened peace efforts.
The Goldstone report recommended that its conclusions be sent on to the International Criminal Court (ICC) prosecutor in The Hague if Israel and Hamas do not hold their own credible investigations into allegations of war crimes within six months.
The report accused Israel of war crimes and crimes against humanity.
It also accused Hamas, which has de facto control of Gaza, of war crime violations, but reserved most of its criticism for Israel.
On Thursday, Navi Pillay, the UN human rights chief, endorsed the report, calling for "impartial, independent, prompt and effective investigations" into the alleged war crimes.
Pillay said: "A culture of impunity continues to prevail in the occupied territories and in Israel," Pillay said during the UN Human Rights Council's special debate session on the report on Thursday.
In her speech, Pillay cited concern about the restrictions on Palestinians wishing to enter al-Aqsa and expressed "dismay" about the Israeli blockade of Gaza that she said "severely undermines the rights and welfare of the population there".
On Thursday, Goldstone, a former South African judge, criticised the resolution, saying: "I hope that the council can modify the text."
About 1,400 Palestinians – the majority of them civilians - and 13 Israelis were killed during Israel's three-week war on Gaza, which had the stated aim of stopping rocket attacks by Palestinian fighters from the coastal territory.