After Suarez overcame a 5-1 deficit to tie the second set at 5, Stephens broke serve for a 6-5 lead and scored four straight points to put away the match, ending it with a sharp cross-court forehand.
"Maybe the old me would have kind of freaked out," Stephens said. "I was kind of like, 'OK, it's fine, she's allowed to play well. Just kind of regroup.' I did that. I was happy with that."
The 25-year-old Stephens, ranked No. 3 in the world, won a hard-court title in Miami in March and lost the French Open final to top-ranked Simona Halep. In the quarterfinals, the American will face Anastaija Sevastova of Latvia, a 6-3, 7-6 (2) winner over 10th-seeded Julian Goerges of Germany.
Sevastova, ranked 19th, won last month in Bucharest.
The top-seeded Halep advanced with a 6-2, 6-2 victory over 13th-seeded Venus Williams in the second match of the day for both players. Halep set up a quarterfinal against sixth-seeded Caroline Garcia of France, a 6-3, 6-2 winner over Russia's Maria Sharapova.
Ashleigh Barty of Australia beat France's Alize Cornet 7-6 (3), 6-4. The 15th-seeded Barty will face Kiki Bertens of the Netherlands, a 6-3, 6-2 winner over eighth-seeded Petra Kvitova of the Czech Republic.
Elina Svitolina of Ukraine, the fifth-seeded defending champion, beat Johanna Konta of Britain 6-3, 6-4 to set up a quarterfinal against 14th-seeded Elise Mertens of Belgium.
Mertens outlasted Aryna Sabalenka of Belarus 2-6, 7-6 (1), 6-0.
In delayed second-round matches, Halep topped Anastasia Pavlyuchenkova 7-6 (9), 4-6, 7-5, Sabalenka upset second-seeded Caroline Wozniacki of Denmark 5-7, 6-2, 7-6 (4), Konta beat Victoria Azarenka 6-3, 6-1, and Williams edged Sorana Cirstea 7-6 (4), 6-4.