International Committee of the Red Cross (ICRC) president Jakob Kellenberger held a 90-minute meeting with Mr Lavrov in Moscow yesterday. He asked Russia to help persuade Damascus to allow more access for humanitarian aid to civilians trapped in areas under fire from President Bashar al Assad's forces.
"It was a positive meeting," ICRC spokesman Hicham Hassan said. "The ICRC received positive indications of support on its operational priorities and its initiative of a two-hour cessation of fighting on a daily basis."
The independent agency hoped to see "concrete results" on the ground in Syria in the coming days or weeks. "The meeting was also focused on the importance today of neutral and independent humanitarian action in Syria," Mr Hassan added.
Russia is one of Mr Assad's few remaining allies and is seen as retaining some leverage on the Damascus Government.
On February 21, the ICRC, the only international agency to deploy aid workers in Syria, proposed a daily humanitarian ceasefire of two hours to allow time to evacuate the wounded safely and deliver food, medicines and other vital supplies.
Meanwhile, rebels fought security forces in Damascus yesterday in the most violent gunbattles the Syrian capital has seen since the start of the year-long uprising against Mr Assad, opposition activists said.
The United Nations says more than 8000 people have been killed and 230,000 forced to flee their home, mostly uprooted within Syria but including at least 30,000 who have moved abroad.
A joint team of experts from the UN and Organisation for Islamic Co-operation, led by the Syrian Government, have started a mission to assess humanitarian needs across Syria.