The influential Republican senator, one of Donald Trump’s staunchest allies in Congress, intervened after Democratic senator Robert Menendez asked for consent to pass the resolution.
The measure, which was passed by the House last month in an overwhelming 405-11 vote, would have provided “official recognition and remembrance” of what many experts consider genocide committed by the Ottoman Turks during the First World War.
"The United States foreign policy must reflect an honest accounting of human rights abuses, crimes against humanity, ethnic cleansing and genocide. We cannot turn our backs on the Armenian victims of genocide," Mr Menendez said on Wednesday.
Mr Erdogan, who earlier attended a meeting with Mr Trump, said during a press conference that Turkey was “hurt deeply” by the House resolution and that it had the potential to cast a “deep shadow over our bilateral relations”.
Following his meeting with Mr Erdogan, Mr Graham objected to passing the resolution in the Senate because senators should not “sugar coat history or try to rewrite it”. Under Senate rules, any senator can ask for consent to pass a resolution, and any senator can block it.
Turkey is also angered at the US for supporting the Kurdish forces it views as a threat and for refusing to extradite a Muslim cleric it accuses of fomenting a 2016 coup attempt against Mr Erdogan.
Mr Graham acknowledged Turkish issues regarding certain Kurdish elements of the Syrian Democratic Forces that partnered with the US, but said they should be addressed by creating a safe zone and not with a "disruptive" incursion that "must end”.
Mr Trump said he and Mr Erdogan are "very good friends”, but their meeting at the White House failed to resolve an issue over Turkey’s decision earlier this year to accept delivery of a Russian air defence system that Nato considers a threat to its security.
The Turkish president told reporters he might be persuaded to use the US-made Patriot system "as well" as the Russian S-400. Mr Trump said they would agree to keep working on the issue.
"The acquisition of the S-400 creates some very serious challenges for us," Mr Trump said. "Hopefully we'll be able to resolve that situation."
The dispute over the competing air defencee systems is a major component of the tension between the two countries. Turkey has also come under fire on Capitol Hill for its incursion into Syria last month to attack the Kurdish forces that fought with the US against the Isis.
Mr Erdogan used the meeting as a chance to defend his military offensive against US-backed Kurdish forces in northeastern Syria, some of whom have links to the separatists who have waged a violent campaign in Turkey for decades.
"We're just fighting terrorists, period," he said. "If you don't fight back, you will have to pay a very hefty price."
His words failed to placate members of Congress and others who accuse Turkish-backed forces of killing Kurdish civilians and causing a humanitarian crisis in the incursion, which prompted the US last month to hurriedly evacuate a small number of American troops from near the Syria-Turkey border.
"There has been a callous disregard for civilian lives, including attacks on residential areas," said Margaret Huang, executive director of Amnesty International USA.
Senate minority leader Chuck Schumer said it was "mindboggling" and "appalling" for Mr Trump to roll out a red carpet for Mr Erdogan after the invasion of Syria.
"Erdogan suppresses free speech, arrests opponents and does so many other terribly things to his country, which was once a much more shining example of democracy," Mr Schumer said.
In the Senate, two Democrats have introduced legislation denouncing Turkey's targeting of journalists, political opponents, dissidents, minorities and others.
They said the Turkish government had imprisoned more than 80,000 Turkish citizens, closed more than 1,500 non-governmental organisations on terrorism-related grounds and dismissed or suspended more than 130,000 civil servants from their jobs.
During the news conference, Mr Trump joked that Mr Erdogan should only call on a friendly Turkish reporter. When Mr Erdogan called on a female Turkish reporter, according to several people in the room, Mr Graham leaned over and quietly told someone seated next to him, "She's the only reporter left over there" in Turkey.
Additional reporting by AP