The 10th seed from Belarus was eliminated on Sunday by Gisela Dulko in a 6-1 6-2 defeat.
Fourth seed Maria Sharapova was also in a hurry, beating 68th-ranked Argentine Gisela Dulko 6-0 6-1 in just 58 minutes.
In Buenos Aires, Gisela Dulko beat world number nine Svetlana Kuznetsova 6-4 3-6 6-4 to give Argentina the early advantage over Russia .
Henin went out to Argentina's Gisela Dulko in her first tournament since reaching the final of the Australian Open in her grand slam comeback.
U.S. Open champion Clijsters avoided suffering the same demise as fellow Belgian and former world No. 1 Justine Henin, who lost to Argentina's Gisela Dulko on Friday.
In other first round action, eighth-seeded Australian Samantha Stosur edged past Argentine Gisela Dulko 7-6 7-5 and 16th seed Nadia Petrova beat Russian compatriot Elena Vesnina 4-6 6-2 6-3.
The Belgian ace had beaten Slovakian Magdalena Rybarikova in straight sets in her opening round match, but found Dulko a tougher test and slipped to a 6-2 1-6 6-4 defeat.
Belarussian Azarenka reached the Paris quarterfinals last year, but has been suffering from a succession of injuries recently and was blown away 6-1 6-2 by the dangerous Gisela Dulko of Argentina.
Meanwhile, former world number one Ana Ivanovic continued her disappointing grand slam form since winning the 2008 French Open as she was bundled out by 36th-ranked Gisela Dulko 6-7 7-5 6-4.
Eighth seed Samantha Stosur beat Argentina's Gisela Dulko 7-6 (7-3) 7-5, and the Australian will face Maria Jose Martinez Sanchez on Tuesday, with the Spaniard having a bye after beating Jankovic in Saturday's Italian Open final.
Star attraction Henin, who put up a tremendous fight before losing to eventual champion Serena Williams in Melbourne, is returning to the WTA circuit after a 19-month layoff, and looked rusty in her loss to Dulko.
World number 131 Scheepers had beaten Gisela Dulko, a player ranked 86 places above her, previously in her campaign as she became the first South African to reach the last 16 since Amanda Coetzer in 1997.
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