Under the measure -- drafted by the Arab League and the Western powers -- Assad would be ordered to halt a crackdown that the UN says has killed more than 5,400 people in 10 months. He would also hand power to his deputy.
"We all know that change is coming to Syria. Despite its ruthless tactics, the Assad regime's reign of terror will end," Clinton said.
"The question for us is: how many more innocent civilians will die before this country is able to move forward?"
Qatari Prime Minister Sheikh Hamad bin Jassim Al-Thani, speaking on behalf of the Arab League, said the UN had to stop Assad's "killing machine."
However, Russia, which has said it fears that the real agenda is regime change, declared that the UN body did not have the authority to impose such a resolution.
Moscow's ambassador to the UN, Vitaly Churkin, argued that Syria should "be able to decide for itself" and said the Council "cannot impose the parameters for an internal settlement. It simply does not have the mandate to do so."
However, the tone of the debate was measured and Churkin did say that the Security Council could "play a constructive role."
Speaking to reporters later, he warned against rushing to pressure Syria.
"You can press the accelerator too hard and you find yourself in a ditch," Churkin said.
But he also hinted at compromise, saying: "I think that we can produce a text which would be useful in a way which would help the Arab League to play its key role."
Russia and the Western-Arab alliance appear deeply divided over the call for Assad's speedy departure.
Earlier, Russian Foreign Minister Sergei Lavrov said during a trip to Australia: "Regime change is not our profession."
The resolution, which was officially introduced by Morocco, calls for the formation of a unity government leading to "transparent and free elections."
It stresses that there will be no foreign military intervention in Syria as there was in Libya, helping to topple Moamer Kadhafi.
At the United Nations, Hague said negotiations would continue.
"We will hold discussions with Russia, the other nations, over the next 24 hours to see if we can make progress on this resolution," he said.
In Washington, US intelligence chief James Clapper said the fall of the Assad regime was in any case inevitable.
"I do not see how he can sustain his rule of Syria," Clapper told senators. "I personally believe it's a question of time but that's the issue, it could be a long time."
Syria remained defiant, with UN ambassador Bashar Jaafari saying his country would "stand firm in confronting its enemies."
He accused the Western-Arab alliance of "double standards" and of "fomenting the crisis."
The opposition Syrian National Council meanwhile deplored the international community's lack of "swift action" to protect civilians "by all necessary means," in a statement on Facebook.
The SNC, the most representative group opposed to Assad, reaffirmed the "people's determination to fight for their freedom and dignity," stressing they "will not give up their revolution, whatever the sacrifices."
The head of the now-defunct Arab League observer mission to Syria, General Mohammed Ahmed Mustafa al-Dabi, said there had been a marked upsurge in violence in the past week.
On Monday alone, almost 100 people, including 55 civilians, were killed during a regime assault on the flashpoint city of Homs, said the Syrian Observatory for Human Rights.
On Tuesday, at least 22 people were killed, all but one of them civilians, the Observatory said.
The rebel Free Syrian Army said half of the country was now effectively a no-go zone for the security forces.
"Fifty percent of Syrian territory is no longer under the control of the regime," its Turkey-based commander Colonel Riyadh al-Asaad told AFP.
He said the morale of government troops was extremely low. "That's why they are bombing indiscriminately, killing men, women and children," he said.
CIA director David Petraeus told senators in Washington that Assad now faced challenges in Damascus and Aleppo, two cities that had been seen as insulated from the unrest.
"I think it has shown indeed how substantial the opposition to the regime is and how it is in fact growing and how increasing areas are becoming beyond the reach of the regime security forces," Petraeus said.