Berdych stopped local hope Somdev Devvarman 6-3, 6-1, while Cilic rallied past Sergiy Stakhovsky of Ukraine 4-6, 6-2, 6-3.
Also, fifth-seeded Benoit Paire of France cruised into the last eight with a 6-1, 6-0 win against Dudi Sela of Israel, while Roberto Bautista Agut of Spain defeated Matthias Bachinger 7-6 (4), 6-3.
Berdych, who had a bye in the first round, dominated 2009 finalist Devvarman and didn't face a break point.
Berdych, who will play Agut in the next round, said his excellent results last season made him work harder in the offseason. In 2012, he won two titles, reached the U.S. Open semifinals and helped his country win the Davis Cup.