Cobain played the retro acoustic-electric 1959 Martin D-18E during a legendary MTV Unplugged performance in 1993, just five months before he died.
At $6.01m after fees, the guitar is the most expensive ever sold at auction, Julien's Auctions said.
Bidding in Los Angeles opened at $1m and was won by Rode Microphones founder Peter Freedman.
Nirvana's acoustic performance for the MTV Unplugged series on 18 November 1993 is now considered one of history's greatest live performances.
Mr Freedman said he planned to take the guitar on a worldwide tour, with proceeds to go to supporting the performing arts.
He described buying the guitar as "a once-in-a-lifetime opportunity", according to the auction house.
The guitar came with its case, which Cobain had decorated with a poster from the band Poison Idea's 1990 album Feel the Darkness.
Before this, the most expensive guitar sold in history was a Fender Stratocaster used by Pink Floyd guitarist David Gilmour. It fetched almost $4m at a charity sale in June 2019.