António Guterres, the agency's high commissioner, made the prediction Sunday in Ankara, Turkey.
Earlier this week, the number of Syrian refugees topped one million people, the agency announced Wednesday. This included Syrians either registered as refugees or being assisted in the same capacity.
In December, the UN estimated the number of Syrian refugees would reach 1.1 million by the end of June, but decided to adjust that estimate based on the present trajectory.
About half of the refugees are children, according to the agency. The majority of the kids are younger than 11.
'This tragedy has to be stopped'
"With a million people in flight, millions more displaced internally, and thousands of people continuing to cross the border every day, Syria is spiralling towards full-scale disaster," Guterres said in a statement. He said the agency is doing everything it can to help, but their resources are dangerously stretched.
"This tragedy has to be stopped," he said.
Guterres echoed this sentiment on Sunday, saying that the agency can mitigate the suffering, but the solution to Syria's refugee crisis is political.
He said the international community should work toward ending the conflict. There is a "risk of an explosion" throughout the already volatile Middle East region if Syria's civil war continues much longer, he said.
The agency says most of the refugees have fled to Lebanon, Jordan, Turkey, Iraq and Egypt. Although, Syrians are increasingly choosing North Africa and Europe as well.
Guterres also renewed a call for nations to help support Turkey, Lebanon, Jordan and other countries shelter the refugees.
Syria's civil war started with anti-regime protests in March 2011. It has since spiraled into a civil war pitting President Bashar al-Assad's forces against hundreds of rebels groups.
Assad's government portrays the conflict as a war fueled by an international conspiracy carried out by terrorists that seeks to weaken the country.
Since the war began, more than 70,000 people have been killed in the violence, according to the UN.