Belarussian Azarenka was beaten 6-2 7-6 by Slovakian 15th seed Dominika Cibulkova and must wait to see if Russian Maria Sharapova will claim the top spot.
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 either U.S. Open champion Samantha Stosur of Australia or American teenager Sloane Stephens in the quarter-finals.
In Image: Victoria Azarenka of Belarus reacts after missing a return in her fourth round match against Dominika Cibulkova