Repeated efforts by the Arab League and Russia to broker talks have been rejected by the Syrian opposition, which refuses any negotiations while the crackdown continues. The opposition has also said Assad's proposed reforms, including a new constitution and eventual multiparty elections, are aimed at keeping his hold on power.
In Tuesday's talks, Assad told Lavrov that Russia's position has played "a key role in saving our motherland," according to ITAR-Tass.
As Lavrov's convoy snaked its way along Damascus' Mazzeh Boulevard, it was greeted by a sea of Assad supporters cheering the vetoes at the U.N.
"Thank you Russia and China," read one banner that had photos of Assad and the Russian president. Many stood in the rain carrying Syrian flags as well as the red, blue and white Russian banner.
"I am here to thank Russia for its stand in the face of the world conspiracy against Syria," said Manya Abbad, 45. "I wish the Arabs adopted similar stances."
The Assad regime says terrorists acting out a foreign conspiracy to destabilize the country are behind the uprising, not people seeking to transform the authoritarian regime.
But in the flashpoints of the conflict, witnesses, residents and human rights workers say Assad's forces are shelling and firing indiscriminately. On Tuesday, the troops renewed their assault on one of the main centers of the opposition, the city of Homs, with activists saying tanks were closing in on a restive neighborhood.
Despite the assault, members of the rebel Free Syrian Army pledged to protect the besieged Baba Amr neighborhood.
"We are just here to respond and defend the local residents from Assad's army snipers," said one fighter, according to Associated Press television footage.
Shielded in the corridors of a deserted building once occupied by Assad's forces, the rebels moved carefully from one position to another overlooking suspected sniper hide outs.
At a makeshift medical clinic, the dead were wrapped in white sheets and piled on a pickup truck outside. Doctors appeared overwhelmed by the number of wounded and the severity of their injuries.
"Can someone help, please!" wailed a man kneeling by a wounded relative on the floor, "Someone come and see him!"
Activists said at least 15 people were killed in violence around the country Tuesday.
Homs was the site of the deadliest assault of the uprising on Saturday, when activists reported more than 200 people were killed in an overnight bombardment hours before the U.N. vote. The government denied the deaths.
Syria has blocked access to trouble spots and prevented most independent reporting, making it nearly impossible to verify accounts from either side.
In Washington, White House spokesman Jay Carney said the U.S. was not considering arming opposition groups in Syria, despite calls from some U.S. lawmakers to consider such an option.
Carney said current deliberations inside the administration were focused on how the U.S. could provide humanitarian aid to the Syrian people and ratchet up pressure on the Syrian government.
U.S. senators John McCain and Joe Lieberman urged the U.S. to explore the prospect of arming opposition forces. "It's an option that now should be on the table," McCain said.
McCain conceded the situation in Syria was more complicated than in Libya, where opponents of Moammar Gadhafi quickly gained control of an eastern city, but he insisted it was necessary.
"I feel very strongly that what's happening in Syria is exactly what we got into Libya to stop Gadhafi from doing," said Lieberman. "The question is what do you do. One of the things is giving support to the Syrian Free Army."
Associated Press writers Albert Aji in Damascus, Zeina Karam in Beirut, Jim Heintz in Moscow, Donna Cassata in Washington and Selcan Hacaoglu in Ankara, Turkey, contributed to this report.