The 35-year-old performer - real name Charlyn "Chan" Marshall - beat records by Regina Spektor and Tom Waits to win the prestigious award.
Founded in 2001, it honours critically acclaimed releases in the US that have not broken into the mainstream market.
Previous winners include Damien Rice, Sigur Ros and rock band N.E.R.D.
This year's panel of judges or "listmakers" included Franz Ferdinand, KT Tunstall and members of Snow Patrol.
'Beguiling voice'
Last year's winner, singer Sufjan Stevens, was also one of the judges.
In a statement, Tunstall called The Greatest "an immediate classic that will never age".
"Cat Power's album quickly got under my skin," said the Scottish singer. "She has one of the most beguiling voices around."
A previous Shortlist finalist for her 2003 album You Are Free, Power is the first female artist to be given the prize.
She received a Brit nomination for best international female earlier this year.