The top-seeded Belarusian was beaten 6-2 7-6 by Slovakian 15th seed
Sharapova, who meets unseeded Czech Klara Zakopalova in the last 16 on Monday and has yet to drop a set here, must get to the Roland Garros final for the first time to become number one.
Cibulkova, who reached the French semi-finals three years ago and had lost seven times to Azarenka in eight previous meetings, collapsed on to her back, a big grin on her face, after winning the tiebreak 7-4 with a backhand crosscourt on her second matchpoint.
The Slovakian will play US Open champion
Stosur had to play on Court One, after men's matches on the main showcourts ran longer than expected, and took her time to finish off her win.
She was broken as she served for the match at 5-3, perhaps disturbed by having to replay a point when a pigeon swooped along the net, but broke Stephens to love in the next game.
Asked how she would recover from the defeat, a stone-faced Azarenka said sarcastically: "I'm going to kill myself!" Analysing what went wrong, she said: "Pretty much everything, really. I don't know how to describe my performance today...I wasn't satisfied being out there playing that way but I guess it happens."
"I am curious to see how far I can go, what level I can get to," said clay specialist Errani, who had knocked out Serbian Ana Ivanovic, the 2008 winner, in the previous round.
Sunday's matches were played under cloudy skies and Kuznetsova, who won in 2009, said the gloomy weather had affected her.
"Today the weather was so bad. I felt cold during the whole match. For me to move was really complicated, I couldn't make my feet move," said the Russian who started working with Marat Safin's old coach Hernan Gumy a week before the tournament.
Errani will now play another first-time quarter-finalist here, 10th seed
The Italian said she would change her tactics to play left-hander Kerber but, pressed on what she would do, told reporters with a smile: "I am not telling you for now, I am keeping it to myself."