The Spaniard appeared to be easing past her Dutch opponent when she took the first set, but Bertens battled hard to take the second to a tie-break.
Suarez Novarro's class showed, though, and she came out on top to seal her place in the last eight.
Third seed Klara Zakopalova progressed as she cantered to a 6-3 6-3 victory over France's Ailze Lim and will next face Alexandra Dulgheru who beat home favourite Teliana Pereira 7-5 6-2.
Dulgheru will be one of two Romanians in the last eight with fifth seed Monica Niculescu following her lead courtesy of a 6-3 6-1 win over Danka Kovinic.
In Wednesday's other last-16 clash, Belguim's Alison Van Uytvanck beat eighth seed Paula Ormaechea 6-2 6-2 to set up a meeting with Spaniard Garbine Muguruza.