The 20th-seeded Vinci, who reached the third round of the US Open last year for her best Flushing Meadows result in nine previous tries, took the unexpected step to the last eight with a forehand passing shot winner.
Vinci outplayed Radwanska with 29 winners, and made just one unforced error from the backhand side to improve on her previous best grand slam showing, a fourth-round result at this year's Wimbledon.
The world number 19 advanced to a quarter-finals showdown against doubles partner and compatriot Sara Errani, who enjoyed an upset 7-6 6-3 win over sixth-seeded Angelique Kerber of Germany in her last 16 16 tilt on Monday.
"I'm happy to play against her," Vinci told the Armstrong Stadium crowd after her victory. "She's my best friend."