Klizan needed three sets to get by Matteo Viola 4-6, 7-6 (8-6), 6-1. Klizan fought off nine of 10 break points in the third set and pulled off four breaks in the set.
He won 63 percent of the points in the set off Viola's serve after taking about 40 percent of the points when receiving serve in the first two sets.
Dimitrov lost a match in which he didn't see a break point. However, he didn't have a break opportunity, either, as Ivo Karlovic came away a 7-6 (7-3), 7-6 (7-3) winner over Dimitrov.
Servers won more than 79 percent of the points in the match. Karlovic edged ahead in the first-set tiebreaker and went up 6-0 in the second in advancing.
Grega Zemlja stopped Ilija Bozoljac 6-7 (5-7), 7-6 (7-4), 6-2 and earned a second-round berth against top-seeded Marin Cilic.
Philipp Petzschner's 7-5, 6-2 win over Mate Pavic put him into a match against No. 2-seeded Andreas Seppi.
Other winners Tuesday were Blaz Kavcic, Igor Sijsling, Lukas Rosol and Somdev Devvarman.