Lavrov made the comments in Qatar's capital, Doha, on Wednesday, at the end of a three-nation tour of the region that also included stops in Kuwait and the United Arab Emirates (UAE).
"We welcome all initiatives to resolve the Gulf crisis, and we support the Kuwaiti efforts in that direction," Lavrov said during a joint press conference with his Qatari counterpart, Sheikh Mohammad bin Abdelrahman Al Thani.
Kuwait has been acting as a mediator since June 5, when Saudi Arabia, the United Arab Emirates (UAE), Bahrain and Egypt announced they were severing relations with Qatar and imposing a land, sea and air blockade against it, accusing Doha of supporting "terrorism".
The four countries have not provided any evidence for their claim, and Qatar has repeatedly denied the allegations as baseless.
Lavrov, who will visit Saudi Arabia and Jordan in September, said the only way to resolve the crisis is through negotiations and dialogue between all parties.
For his part, Qatar's foreign minister said that his country was the only side in the dispute that responded to Kuwait's efforts positively.
"Our goal is to resolve this conflict and end the blockade, not just sitting at the table for the sake of negotiations only," he said.
Ibrahim Fraihat, a Qatar-based political analyst, described Lavrov's visit to Qatar as a "major development" in the Gulf crisis and an attempt by Russia to increase its political influence in the oil-rich region, which has traditionally been under Washington's political and military umbrella.
Fraihat argued Russia was adding its weight behind Kuwait's mediation efforts in a bid to fill a regional vacuum created by the United States' lack of clear policy in helping to end the dispute.
"Russia is stepping in because the Americans have failed to exercise their historic political leverage with all parties in order to bring the parties to the negotiation table," added Fraihat, a professor at the Center of Conflict and Humanitarian Studies in Doha.
"Russia enjoys excellent and diverse ties with all the parties," he said, adding that Lavrov's visit "may have major implications in the future".
Lavrov, who also met with the Qatar's Emir Sheikh Tamim Bin Hamad Al Thani, said his trip was aimed at enhancing the economic and political ties between the two countries.
He said his country is looking forward for more cooperation with Qatar in the areas of economic investments, especially in the oil and gas sector.
The talks also focused on a host of other regional issues, including the Israeli occupation of the Palestinian territories, according to Qatar's foreign minister.
Both foreign ministers called for unity between the Palestinian factions of Hamas and Fatah and the need to resolve the conflict based on the Arab peace initiative and the two-state solution.