Mattek-Sands, seeded fifth, trailed 4-1 and was 15-0 down in the third set but suddenly turned a corner as she won 20 of the next 22 points on her way to a 5-7 6-3 6-4 win.
“Mandy played very well today. She was hitting the forehand very well,” Mattek-Sands said on the WTA website.
“When she was up in the third set I really had to fight hard and play well to come back.”
However, Chirico, seeded sixth, lost 6-1 6-4 to Anastasija Sevastova of Latvia.
The other seeds in action all progressed. Annika Beck, seeded third, easily saw off Brazilian wildcard Carolina Alves 6-1 6-2, while Brazilian number one Teliana Pereira, seeded fourth here, took a 6-7 (3/7) 6-3 7-5 win over Maria Irigoyen.
Seventh seed Laura Siegemund saw off Marina Melnikova 6-2 6-1.
There were also wins for Maria-Teresa Torro-Flor, Veronica Cepede Royg, Risa Ozaki, Elitsa Kostova, Gabriela Ce, Ana Bogdan and Tereza Martincova as the first round wrapped up.