Bogomolov — who was making his debut at a second consecutive event in Russia after reaching the quarterfinals at the Kremlin Cup last week — broke Kunitsyn twice in the first set and three times in the second.
"I know Kunitsyn and the way he plays perfectly," Bogomolov said. "It was important to neutralize his backhand. So, I served hard and returned with maximum concentration."
The 22nd-ranked Cilic is looking for his first title of the season. Cilic broke in the ninth game of the first set but Bubka saved five break points in the 11th game of the second set. He won in a rout in the tiebreaker.
"It was really good for me to go through this match today," said Cilic, who reached the semifinals of the event four years ago. "Bubka was serving really good in some moments and especially in the second set. On 5-all I had five break points and he served three aces and one service winner, so, it was really tough."