The 30-year-old Frenchman restricted the big-serving Karlovic, seeded seventh, to nine aces and came up with nine of his own to win in an hour 31 minutes.
Fourth seed Cilic, the 2014 U.S. Open champion, struggled on serve but was still too strong for Monaco and the Croatian next faces Belgian David Goffin, who secured a 1-6 7-5 6-2 victory over Joao Sousa of Portugal.
"My serve was a bit up-and-down today. It's something I'll definitely have to straighten out for my match tomorrow," Cilic told the ATP website (www.atpworldtour.com).
"He returned really well in the first set. In the second after I broke him, I could see that he was having some trouble. It's unfortunate for him since I know he's had a lot of injuries."
Australian Kyrgios, who is yet to lose his serve this week, continued his fine run, firing down 18 aces against Luxembourg's Muller.
"I thought I played great today. I served well, returned well and my engagement level was really good the whole time," Kyrgios said.
(Reporting by Ian Rodricks in Bengaluru, editing by Ed Osmond)
