Starr used the Ludwig kit in more than 200 performances after taking possession of it on 12 May, 1963.
Last month, Julien's Auctions sold a drum head from a kit Starr played on US television show The Ed Sullivan Show in 1964, also for $2.1m.
The ex-Beatles drummer has put several memorabilia items up for sale.
Also available is a guitar used by John Lennon, which he gave to Starr in 1968, some of the drummer's signature rings and his own copy of The Beatles' White Album, marked as number 0000001.
Part of the sale proceeds will go to the Lotus Foundation, founded by Starr and his wife Barbara Bach, which funds and promotes charitable projects aimed at advancing social welfare.
According to the AP news agency, the Ludwig drum kit was sold to Jim Irsay, owner of the American Football team Indianapolis Colts.
The auction runs from until Saturday in Beverly Hills and also online via Julien Auction's website.
A drum kit played by Ringo Starr on Beatles hits including Can't Buy Me Love and I Want to Hold Your Hand has sold at auction for $2.1m (£1.4m).