Putin lauded both players for "an outstanding game."
India's Anand beat Gelfand of Israel on Wednesday 2.5-1.5 in a rapid tiebreaker following a 6-6 draw played at the Tretyakov Gallery, one of Russia's finest museums.
Gelfand, who was born in the Soviet Union and emigrated from Belarus to Israel in 1998, was cheered throughout the match as if he were a home player.
Anand and Gelfand join Russian president for tea (Photo: AFP)
Anand told Putin over tea Thursday that he had learned to play chess in a Soviet cultural center in his hometown when he was a child.
"So we brought this on ourselves!" Putin said.
Putin is scheduled to visit Israel at the end of June, following talks between high-level officials in Moscow and Jerusalem.
President Shimon Peres will host a special ceremony saluting Russia's role in liberating the death camps at the end of World War II.