Cilic was broken early in the first set and he struggled to recover as he went on to lose 6-4.
But the Croatian responded in the second with a clinical display right when it mattered the most.
Cilic, who is the No 1 seed following the withdrawal of Rafael Nadal and Juan Martin del Potro, then went on to secure the victory with an identical 6-3 final-set win.
Cilic will face American Sam Querrey in the quarter-finals and he was delighted to get through as he takes another step towards avenging last year’s final defeat.
“It was difficult,” Cilic said.
“I didn’t serve that well. I was break down, and it was extremely difficult to dig that [first] set back.
“He served really, really good consistently, and very tricky with left hand.
“He was hitting good spots everywhere.
“It was definitely not easy but I upped my serve from beginning of the second set.
“I didn’t give him many chances on the return and that just pushed me to stay in the match and to put back pressure on him.
“Obviously that gave me few more chances on the returns.
“I got that break, and sort of things opened up a little bit more for me.
“I was quite solid. Really happy with the way I pulled it from one set down.”