The 26-year-old Belarusian had been the highest seed left in the women's draw after Agnieszka Radwanska, Angelique Kerber and Garbine Muguruza all lost.
"Unfortunately in my first match I think I tweaked my back a little bit. It was really cold, and, I don't know, I think it's much easier to get a little bit hurt when it's that cold," Azarenka told reporters.
"The second match obviously I was in a lot of discomfort. Having a day off I thought it was going to get better. Today before my warm-up it did feel better, but once I started hitting it came back."
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Azarenka, who beat Britain's Laura Robson in her opening game and has twice been a finalist in Madrid, said she hoped to recover in time to play in Rome next week, her last claycourt event before the French Open.
Qualifier Chirico, at 19 the youngest player left in the tournament, will face Australia's Daria Gavrilova in the quarter-finals.
Gavrilova upset defending champion and fifth seed Petra Kvitova 6-3 6-4.
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