US Ambassador Nikki Haley told the assembly before the vote that it could make history and unconditionally speak out against Hamas, which she called "one of the most obvious and grotesque cases of terrorism in the world".
Israeli Ambassador Danny Danon lamented after the vote that a decision to condemn Hamas had been "hijacked" by a procedural motion that required adoption by a two-thirds majority, instead of a simple majority.
It was the first proposed resolution condemning Hamas to be presented to the 193-nation assembly, which has been meeting since 1946.
"There is nothing more anti-Semitic than saying we can not condemn terrorism against Israel, while we would not hesitate for a minute to condemn the same acts if they were taken against any other country", she went on to say, noting the thousands of rockets from Hamas fired at civilian areas in Israel.
It was the first time a United Nations resolution specifically targeted Hamas.
Both the United States and European Union have classified Hamas as a terrorist group.
The vote was symbolically important because, from the time Haley arrived at the U.N.in January 2017, she has made a point of trying to turn the tide of what the Trump administration considers overwhelmingly anti-Israel resolutions and this vote illustrated what an uphill battle that turned out to be.
Palestinians had initially sought to thwart the U.S. move by presenting an amendment to the USA text to include a reference to United Nations resolutions that condemn Israeli settlements, call for negotiations on East Jerusalem and pledge support for the two-state solution.
Haley called condemnation of Hamas "an essential step" to a peace settlement.
US officials had been hopeful that the resolution would pass, but the adoption of the two-thirds threshold proved critical.
Nabil Abu Rudeinah, Abbas's spokesman, said in a statement that the vote against the resolution "is a message to the American administration and Israel that all the conspiracies against the Palestinian leadership will not come to pass".
The Palestinians have severed ties with the Trump administration since the decision a year ago to move the USA embassy to Jerusalem and declare the city Israel's capital. "The reality can not be avoided any longer: the UN is the headquarters of modern antisemitism and the people of the United States should rescind their support accordingly".
Tel Aviv tried to undermine the defeat, with Israel's United Nations envoy Danny Danon saying that a majority vote was within reach had the vote "not been hijacked by a political move of procedure". He also blasted those who voted against the measure. "Your silence in the face of evil reveals your true colors". Germany voted in favor of the draft resolution.
The assembly also adopted by a wide margin of 156 to six with 12 abstentions a Palestinian-drafted measure, presented by Ireland, calling "for the achievement, without delay, of a comprehensive, just and lasting peace in the Middle East" based on United Nations resolutions.
Haley had been working with the European Union and other regional blocs to get the support for the US resolution, which it did, Falk said.
In the letter, seen by Fox News, she warned of potential attempts to disrupt the resolution with amendments by other countries, and urges members to vote against any amendments "or other efforts to undermine adoption of the text".
The U.S. withdrew from the U.N. Human Rights Council earlier this year, citing complaints about an alleged anti-Israel bias of the global body.
Despite Riyadh's close ties to Washington, Saudi Arabia's ambassador to the UN Abdullah Al-Mouallimi urged member states to reject the USA measure.
Thursday's resolution does not only target Hamas, he said, but rather multinational support for the Palestinian people.