Neither player faced a break point in the first set, but Muller saved five in the second.
Cilic had two match points at 6-4 in the tiebreak, then saved a set point at 7-6 down as Muller fought back strongly.
"I was expecting a close match," said Cilic, runner-up here to Tomas Berdych two years ago. "Luckily, I was serving well, and I was mentally focused. Gilles is a very difficult player to play against indoors, and I'm happy to be through."
Kohlschreiber proved clinical on Dodig's serve, converting four of five break-point chances, and was solid on his own, winning 75 percent of points on second serve.
Monfils served for the match at 5-4 in the second set, then lost three straight games.
He finally served out the victory to set up a match against either countryman Gilles Simon, seeded third, or German teenager Alexander Zverev. Simon and Zverev each won their first-round matches on Wednesday.