Brown, ranked 84th in the world, broke serve against Cilic once in each set and did not face a break point en route to his upset victory. He wrapped up his victory with an ace.
The German will face either fifth-seeded Feliciano Lopez of Spain or France's Benoit Paire in the quarterfinals.
The seventh-ranked Cilic also was ousted in the second round at the Australian Open.
Qualifier Kenny De Schepper of France upset sixth-seeded Mischa Zverev of Germany 6-4, 6-3 in the opening round. De Schepper, ranked 162nd in the world, had not won on the main circuit since his round-one victory last year in Montpelier.
Other first-round action saw fifth-seeded Feliciano Lopez of Spain top France's Julien Benneteau 6-3, 4-6, 7-6 (7-0), and Tunisia's Malek Jaziri rallied past France's Tristan Lamasine 1-6, 7-5, 7-6 (7-3).
Frenchman Paul-Henri Mathieu bested countryman Vincent Millot 6-2, 4-6, 6-3 while France's Pierre-Hugues Herbert breezed past countryman Calvin Hemery 6-3, 7-5 to conclude first-round play.