Connect to share and comment
The 25-year-old Sharapova, the No. 2 seed, had a fight on her hands in the first set with Suarez, who used a strong return game and solid one-handed backhand to stay in the match at the Indian Wells Tennis Garden on Sunday.
Sharapova came out charging in the second set and overpowered the 24-year-old Suarez.
The Russian star will play Spain's Lara Arruabarrena-Vecino in the Round of 16.
Arruabarrena-Vecino upset Italy's Roberta Vinci 2-6, 6-4, 6-4 in a marathon match that went two hours and 44 minutes.
In other action on the women's side at Indian Wells, Klara Zakopalova of the Czech Republic upended Dominika Cibulkova of Slovakia 6-4, 7-5.
On the men's side, world No. 1 Novak Djokovic of Serbia defeated Italy's Fabio Fognini 6-0, 5-7, 6-2, while France's Jo-Wilfried Tsonga beat American veteran James Blake 7-6 (8-6), 6-4.