The plane was forced to take a longer route over Greece to deliver Lavrov to a meeting of foreign ministers of the Organisation for Security and Cooperation in Europe (OSCE) in Skopje, North Macedonia.
Kremlin spokesman Dmitry Peskov called Bulgaria's position "absurd and stupid".
Russian Foreign Ministry spokeswoman Maria Zakharova, who is under European Union sanctions, said her presence on board the plane was the reason given by Bulgaria's foreign ministry for denying access to its airspace.
No comment was immediately available from the Bulgarian ministry.
Zakharova wrote on the Telegram messenger app: "The malicious stupidity of the Russophobes reached the point that, for the first time in our history, official authorities banned not an airplane, but a person on that airplane."
Zakharova suggested that Russia could apply similar overflight bans to "thousands of NATO functionaries", and accused Bulgaria of creating "a dangerous precedent".
Ukraine, Estonia, Latvia and Lithuania declared a boycott of the meeting in Skopje in protest at the Russian delegation's presence.
(Reporting by Felix Light; editing by Mark Trevelyan and Kevin Liffey)
