Wozniacki was 5-3 down in the second set when a Cibulkova shot was wrongly called out.
Umpire Emmanuel Joseph overruled the call and awarded the point to Cibulkova instead of replaying the point even though Wozniacki had clearly returned the ball in court Thursday.
That prompted a lengthy dispute between the Danish player and the umpire. Despite not getting the point, Wozniacki seemed fired up by the controversy and saved four match points and broke Cibulkova's serve before Cibulkova eventually closed out the match in straight sets.
"I was aggressive. Even if I did some mistakes I was really confident. It was the only way I could beat her," Cibulkova, of Slovakia, said in televised comments. "I'm happy with the way I stayed really tough mentally."
Bencic was broken four times in a chaotic first set in which she recorded six double faults, including one on break point that allowed Beck to serve for the set at 6-5. But the Swiss recovered to break Beck and dominate both the resulting tiebreak and second set.
"I had some trouble in the start, and I think the court was very slow," the 18-year-old Bencic said in televised comments. "My timing was a little bit off."