The Slovakian, who was beaten in last year's final by Li Na, possessed more of a killer edge on the day, notching up 44 winners to Azarenka's 32, with both players making 29 unforced errors.
Cibulkova was the more aggressive of the pair in the first set, breaking Azarenka three times in the first set, although she surrendered one of those breaks on her way to taking the first set 6-2.
Azarenka noticeably upped her game in the second set, with her first serve significantly better. The Belarusian took the chances that came her way, capitalising on all three break points in a set she won 6-3.
It may have appeared that Azarenka was surging back at just the right time, but Cibulkova was able to arrest her opponent's momentum in a deciding set that lasted nearly an hour, with each game hard fought. It was Cibulkova who came to the fore as the match reached its key moment, with the 25-year-old coming out on top 6-3 in the third set.
Cibulkova will face the winner of the clash between Serena Williams and Garbine Muguruza in the quarter-finals.