In the longest match of the tournament so far, lasting nearly three hours, the second-seeded Croation beat Granollers 6-7 (3-7), 6-3, 6-4.
"He (Granollers) played really well," Cilic said. "He was going on the offensive from the beginning, coming up with some brilliant shots. I found it really difficult to match him."
Serbian Janko Tipsarevic, seeded fourth, also made it through to the quarters with an easy win over wildcard entrant Somdev Devvarman of India.
A losing finalist last year, Devvarman lost 2-6, 1-6 in 72 minutes.
In Wednesday's other matches, Giraldo beat Jan Hajek of the Czech Republic 6-4, 6-2, while Thiemo de Bakker of the Netherlands prevailed over his countryman Robin Haase 6-3, 6-4.
Clijsters wins opening match at BrisbanePrimera Liga round-up
