Cilic had eight aces and broke Kubot four times to wrap up the win in 1 hour, 33 minutes.
American Sam Querrey also advanced with a 6-3, 6-1 victory over Chinese wild-card entry Li Zhe. The 22nd-ranked Querrey jumped out to a quick 5-2 lead in the first set, then won the final five games of the second to close out the match in just 59 minutes.
Querrey had 11 aces and converted five of his 10 break points.
"I thought I served great," Querrey said. "I thought I returned well, and I took advantage of the break points. That was the key."
Grigor Dimitrov of Bulgaria beat Pablo Andujar of Spain 7-5, 6-3 to set up a meeting with second-seeded Novak Djokovic in the next round. Djokovic, who is coming off his third title at the China Open in Beijing, and the other seven top-seeded players have byes in the first round.
Also Monday, Tommy Robredo of Spain defeated qualifier Alex Bogomolov Jr. of Russia 7-6 (4), 6-4; Benoit Paire of France ousted qualifier Philipp Petzschner of Germany 7-6 (2), 6-4; and ninth-seeded Nicolas Almagro of Spain lost to 34-year-old Tommy Haas of Germany 4-6, 7-6 (4) 6-2.