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Azarenka, who missed much of last season after being embroiled in a custody battle over her son, was broken to love in the 12th games of each set to be dumped out by world number 57 Siniakova.
The two-time Australian Open champion has now won just two matches at Roland Garros in the last five years.
Czech Siniakova moves onto a second-round match with Ukraine's Kateryna Kozlova, who stunned defending champion Jelena Ostapenko on Sunday, in a part of the draw in which the only remaining seed is number 26 Barbora Strycova.
It was Azarenka's first Grand Slam match since losing to Simona Halep in the Wimbledon fourth round last July.