Putin said he planned to follow up the meeting with telephone calls to President Donald Trump, and to Middle Eastern leaders including the Emir of Qatar.
The meeting was unannounced and the Kremlin did not make it public until Tuesday morning.
Assad thanked Putin for Russia's help in fighting terrorism and maintaining Syria's territorial integrity, according to the transcript.
US President Donald Trump and his Russian counterpart Vladimir Putin held a lengthy phone call Tuesday, stressing the need to ensure "the stability of a unified Syria", the White House said.
The US and Russian Federation also have wildly different visions for the future of Syria, with the two countries staking out different positions on Assad's future in the country and the involvement of rebel factions.
Putin and his government have been one of the chief supporters of the Syrian President's government, both militarily and in helping negotiate ceasefires in the country's long-running civil war.
In parallel with the UN-backed Geneva process, the Astana talks guaranteed by Russia, Iran and Turkey have led to the establishment of four de-escalation zones in Syria.
The Foreign Ministers of Russia, Iran and Turkey met in the Turkish city of Antalya to discuss the preparations for a Syrian national dialogue congress, the Russian Foreign Ministry said. They agreed on a number of principles for the future of war-torn Syria.
Faced with pressure from other nations urging Mr Assad to step down, Moscow has insisted that it is up to the Syrian people to vote Mr Assad in or out.
"Regarding our joint operation to fight terrorists in Syria, this military operation is indeed coming to an end", he told Mr Assad in televised remarks.