A host of top officials called on the divided UN Security Council to take stronger action to deal with the fallout from the two-year-old conflict, in which up to 100,000 people are believed to have died.
"The extremely high rate of killings nowadays -- approximately 5,000 a month -- demonstrates the drastic deterioration of the conflict," UN assistant secretary general for human rights Ivan Simonovic told a council meeting.
"In Syria today, serious human rights violations, war crimes and crimes against humanity are the rule," Simonovic declared.
Nearly 1.8 million people are now registered with the United Nations in countries around Syria and an average of 6,000 people a day are now fleeing, UN High Commissioner for Refugees Antonio Guterres said.
"We have not seen a refugee outflow escalate at such a frightening rate since the Rwandan genocide almost 20 years ago," Guterres added.
More than two million Rwandans fled the 1994 genocide, in which radical Hutus killed some 800,000 Tutsis and moderate Hutus in a period of about three months.
Guterres said the acceptance of refugees by Lebanon, Iraq, Jordan and other countries was "saving hundreds of thousands of lives."
UN humanitarian chief Valerie Amos said the international community may have to consider cross-border operations to get aid into Syria.
Amos added that $3.1 billion was still needed for operations in and around Syria for the rest of the year.
She said four million people inside Syria need assistance and "considerable restraints" have been imposed on aid agencies by the government and rebel groups.
Amos highlighted the Old City in Homs, where the government has stepped up a siege in the past month. The United Nations estimates that 2,500 civilians are trapped there.
"Opposition groups have so far not enabled them safe passage to leave, and the government of Syria has refused to allow agencies to deliver assistance into the Old City," she said.
Amos appealed for the lifting of bureaucratic obstacles but also the designation of "priority humanitarian routes" and prior notification of military offensives.
Amos said there should be "humanitarian pauses" to allow aid access and "cross-border operations, as appropriate."
The Syrian government is opposed to any cross-border aid, as is Russia, a veto-wielding member of the Security Council and a key international backer of Syrian President Bashar al-Assad.
Turkey, a staunch supporter of the rebels, backed the call for such aid.
"The council needs to consider alternative forms of aid delivery, including cross-border operations," Turkey's deputy UN ambassador Leven Eler said.
Eler said the Syria crisis was turning into "the biggest humanitarian tragedy of the 21st century."
Lebanon's UN ambassador Nawaf Salam told the meeting it was now "urgent" for the Security Council to act on the refugee crisis.
"Increasing cross-border fire and incursions from Syria in Lebanon are threatening the security and stability of my country," he told the 15 ambassadors on the council.
In the latest such incident, mortar fire from inside Syria hit the Israeli-controlled side of the Golan Heights on Tuesday as Syrian rebels and regime forces battled nearby, an AFP correspondent reported.
In a letter of protest, Israel's UN envoy Ron Prosor warned that while Israel had so far shown "maximum restraint" over such incidents, it would "continue to exercise its right to self-defense."
Salam said the UN has registered 607,908 refugees in Lebanon, but the government estimates the true figure at 1.2 million. He estimated the number would grow 20-fold during 2013.
Lebanon has a population of about four million, and he said the influx was the equivalent of 75 million refugees flooding the United States.
Syria's UN ambassador Bashar Jaafari disputed the UN death toll as "unprofessionally sourced" and criticized the use of an American company to collect data.
But Simonovic said that "rigorous" methods had been used to check a death toll of more than 92,900 given one month ago.
He said each death was checked by name and date and cross-checked with at least three sources.
UN leader Ban Ki-moon has since said that up to 100,000 people have been killed in Syria.