No major decisions are expected at the talks over dinner in Geneva, but the Obama administration hopes to improve relations after a post-Cold War low during George W Bush's presidency.
When Russia sent tanks and troops into Georgia last year, the Bush administration sought to isolate Moscow, particularly in international institutions like Nato which suspended ties.
"I don't think you punish Russia by stopping conversations with them," Clinton told reporters in Brussels on Thursday.
"I think that what we have to be is willing to vigorously press the differences that we have, while seeking common ground wherever possible," said Clinton, who has already exchanged letters with Lavrov over a wish to start a new chapter in ties.
In a decision likely to help the atmosphere, Nato agreed on Thursday to resume formal ties with Russia in hopes of securing greater support for the alliance's Afghan military campaign.
The Obama administration has said it wants to "press the reset button" and has focused on Afghanistan, missile defence, nuclear disarmament and Iran as areas for possible movement.
The Kremlin says it is ready to widen cooperation, and Russian officials believe that unless the Taliban insurgency in Afghanistan is contained, Islamist militancy could spread through former Soviet states in Central Asia and reach Russia.
Washington is considering nearly doubling the size of its military presence in Afghanistan and wants to secure alternative supply routes as it faces closure of an air base in Kyrgyzstan and militants have attacked convoys using routes via Pakistan.
The United States and Russia have clashed over a missile shield Washington is planning in Europe to deter any attack from a country like Iran, and Clinton says she wants to get talks with Russia over the issue on a "serious track".
The plan to site missiles and a radar tracking station in former Soviet bloc countries Poland and the Czech Republic has angered Moscow, which sees it as a threat.
US officials say Washington has offered to slow down deployment of the shield in exchange for Moscow's help in curbing Iran's nuclear ambitions.
Obama, who has been lukewarm on missile defence, has denied cutting a deal with Moscow. Russia has said it was willing to talk about the shield but saw Iran's nuclear programme, which the West suspects is for atomic weapons, as a separate issue.
An early area of US-Russian cooperation is likely to be on reducing nuclear weapons and replacing a strategic arms treaty, Start, which is set to expire at the end of this year.
Russia hopes Washington will revive a bilateral civilian nuclear pact, potentially worth billions of dollars in trade, which was withdrawn from the US Congress over Georgia.
Clinton could capitalise on a wave of goodwill that had flowed from Obama's election, said a senior US official.
"The new administration has much more political juice as it leans forward and tries to re-engage the Russians and because of that it will have much more political capital," said the official, asking not to be identified.
Russia might recalculate tactics because of the need for cooperation over the global economic crisis, said the official.
But Russia expert Steve Pifer said domestic pressures could lead to Moscow dragging its feet on improving ties.
"There is always the idea in Russia that there is a foreign enemy and this distracts from domestic woes," said Pifer, a former US ambassador to Ukraine.