World number 13 Cilic raced into a two-set lead, before he was pegged back by his Japanese opponent during a third-set tie-break.
However, the Croatian responded in the fourth set with a string of aces, which allowed him to book his place in the next round.
"I served well. I think that was the key," Cilic told reporters. "He played really well in the tie-breaker. We both had a lot of chances in the match."
Cilic will now take on Martin Klizan for a place in the quarter-final.