The former world No.3 is aiming to become the sixth active player to reach 20 titles on the ATP Tour this week but was staring down the barrel when trailing 5-4 in the deciding set.
But Cilic survived to record a 6-2 2-6 7-5 win against his Korean opponent and earn a last-four date with Alexei Popyrin, after the Australian's straight-sets win over compatriot Matthew Ebden.
"I think that the third set was solid from the both of us, it was a great match and it went to the end," Cilic said. "At 5-5 [in the third set], I kept my focus and felt that I was going to get my chance.
"There were a few break opportunities that I couldn't convert, but on the third one, I hit a great return."
Radu Albot is into a first ATP Tour semi-final in 19 months after knocking out No.1 seed Adrian Mannarino 6-3 6-4. His reward is a semi-final with Alexander Bublik, who overcame No.5 seed Yoshihito Nishioka in straight sets.
At the Open Sud de France, No.1 seed Roberto Bautista Agut made short work of Ugo Humbert in a 6-3 6-3 win. He next faces Peter Gojowczyk after the German routed Dennis Novak 6-4 6-1.
No.2 seed David Goffin cruised past Lorenzo Sonego 6-4 6-4 and will play Egor Gerasimov in the last four after his three-sets win over Alejandro Davidovich Fokina.