"The Syrian people want and deserve change and that should insofar as possible come about through peaceful means," she added, flanked by Swedish Prime Minister Fredrik Reinfeldt and Foreign Minister Carl Bildt. "It must be change that represents the rights and dignity of all Syrians."
The comments represent Clinton's latest salvo in what has been a yearlong diplomatic tussle between Washington and Moscow over the Assad government's crackdown on protesters, who now have taken up an armed rebellion. Activists say as many as 13,000 people have died.
Russia has twice vetoed U.N. Security Council resolutions that would have set world penalties against the Syrian government. The U.S. has chastised its former Cold War foe for continuing to supply weapons to Assad's military and standing by a leader whom many countries have now declared illegitimate.
Still, Clinton hinted that the two countries might be able to come to an understanding on what would in all likelihood have to be a nonmilitary path forward in Syria. She stressed the importance of all the points of U.N. mediator Kofi Annan's peace plan, which includes a call for a political transition, and said Assad's departure "does not have to be a precondition, but it should be an outcome, so the people of Syria have a chance to express themselves.
The tone suggested some flexibility on the timing of a power change. She said Lavrov also spoke of transition in their private talks.
"He himself has referred to the Yemen example," Clinton said, referring to the deal this year that saw longtime dictator Ali Abdullah Saleh step down and his vice president lead a caretaker government. "It took a lot of time and effort with a number of countries who were involved at the table working to achieve a political transition and we would like to see the same occur in Syria."
Clinton, in Sweden as part of a weeklong Europe trip, spoke hours after Assad delivered a televised speech to the Syrian parliament, denying that his government had anything to do with last week's massacre of more than 100 people in the town of Houla, which drew worldwide condemnation.
Assad blamed foreign-backed terrorists and extremists for the bloodshed and pledged to press ahead with his military crackdown.
Clinton condemned the Assad government's "vicious and systematic attacks." With violence spilling over in recent days into neighboring Lebanon, she said she would meet in Turkey on Wednesday with neighbors of Syria who are "anxious about what is happening."
"We could see a full-fledged civil war with consequences that would bring in the rest of the region in ways that could be quite dangerous and are certainly unpredictable."
In Washington, Sen. Richard Lugar said the U.S. should work with Turkey to establish safe zones for retreating Syrian rebels. Lugar, the top Republican on the Senate Foreign Relations Committee, said those areas would be on Turkish territory and guarded by Turks.
Turkey has reacted to the uprising in Syria by accepting some 23,000 Syrian refugees and playing host to civilian and military members of the Syrian opposition. But Turkey has expressed concern about Syria allowing Kurdish rebels who are fighting Turkish forces to establish bases in Syrian territory.