The 16th-seeded Safarova pulled off a 7-6 (11-9), 7-6 (7-5) upset of fourth-seeded Australian Samantha Stosur, who was the runner-up to Serena Williams in last year's event.
Meanwhile, world No. 1 Azarenka decided not to complete a rain-delayed match against Austrian Tamira Paszek. That match had been suspended by rain Thursday with the score tied 3-3 in the first set.
The Belarusian star -- last week's Olympic bronze medalist -- said a left knee injury prompted her decision to retire from the match.
In other action Friday, fifth-seeded Petra Kvitova of the Czech Republic blew out ninth-seeded Marion Bartoli of France 6-1, 6-1, while a match between Canadian Aleksandra Wozniak and Christina McHale was suspended by rain with Wozniak leading 7-6 (7-5), 5-2.
Angelique Kerber and Caroline Wozniacki also posted victories Friday.
Notable deaths of 2014 [PHOTOS]
Michael Sam cut from the Dallas Cowboys