Lavrov spoke after he brokered a meeting between the foreign ministers of both nations. Russia helped secure a truce to halt a six-week conflict in 2020, but the agreement has not led to lasting peace, and armed clashes are common along the border, according to Reuters.
Armenia and Azerbaijan were due to discuss the unblocking of transport links at a meeting next week, where Russia will also be present.
"We hope the outcome will be positive. The parties are already close to a final agreement," Lavrov said in a statement but did not give details.
Armenian Prime Minister Nikol Pashinyan said on Thursday he had agreed to peace talks in Moscow on May 25 with Azeri President Ilham Aliyev and Russian President Vladimir Putin mediating, the Interfax news agency reported.
But during Friday's talks, the two nations confirmed that without progress on settling disputes over borders and transport links, as well as improving the security situation in and around Karabakh, "it is difficult to move forward on specific aspects of the peace treaty", Lavrov said.
MNA/PR