No. 6 seed Feliciano Lopez needed three sets to move to the next round.
Anderson, the No. 4 seed, reached the quarterfinals when Ivo Karlovic retired because of a stomach ailment after Anderson won the first game. Karlovic was a finalist in Memphis last week.
Cilic, the No. 7 seed, made easy work of Benjamin Becker, taking only 55 minutes to win 6-1, 6-3 in a first-round match. Cilic has won 14 of his 18 matches this season.
Lopez rallied from a set down to defeat Adrian Mannarino 4-6, 6-4, 6-2.
The most surprising result of the afternoon occurred in a first-round doubles match when Sekou Bangoura and Vahid Mirzadeh of the United States upset No. 2 seed Eric Butorac and Raven Klaasen 7-5, 3-6, 10-5 after learning only the day before that they were in the tournament as an alternate.
''The other day I was giving lessons, and now I'm here,'' said the 27-year-old Mirzadeh, who lives a short distance away in Wellington, Fla. ''A win like this lifts your spirits and shows that maybe I can compete at this level.''
Bob and Mike Bryan, the No. 1 doubles seed, breezed to a 6-1, 6-3 first-round win against Mannarino and Michael Russell.