In 1741, Kvasov helped Mikhail Zemtsov to prepare coronation celebrations in Moscow.
Two years later, he was entrusted with interior decoration of the Catherine Palace, which resulted in the Grand Ball Hall and other celebrated rooms.
In 1748 he went to the court of the Ukrainian hetman Kirill Razumovsky, Aleksey's brother, to design the residences and churches in Baturin, Glukhov, and Koselets.
The cathedral in Kozelets is his major surviving work in the Ukrainian Baroque idiom.
Kvasov's younger brother Aleksei is credited with devising the general layout of Kazan (1766), Tver (1767), Astrakhan (1768), Kharkov (1768), and the Admiralty part of St Petersburg.