Cuevas will play Fognini, who was one set and a break down before he managed to turn the match.
Fognini started the final set more convincingly and Coric had his last chance at 3-4 but failed to convert break points. Fighting cramp in the ninth game, Coric lost serve again.
"I did not play my best tennis, he was better than me (a) big part of the match," Fognini said. "I was fighting for every point even though I did not feel very well."
Robredo won the first set in only 23 minutes. Carreno Busta was twice a break up in the second set before Robredo responded to win the last three games of the match.
Cilic hit 10 aces and converted five break points to win in just over an hour. The Croat won this tournament in 2012 and reached the 2011 final.
Fifth-seeded Rosol won his first game at the start of the second set.
"I expected his aggressive play and I was prepared," Cilic said. "I did not expect to lose two games only."
Cuevas hit 12 aces for his sixth straight win in Umag, including three qualifying rounds.