Fognini, who received a first-round bye at the clay-court event, saved both break points he faced and broke the 61st-ranked Kazakh player once in each set.
The top-seeded Italian will next face the seventh-seeded Santiago Giraldo of Colombia, who defeated qualifier Philipp Davydenko of Russia 6-1, 6-7 (5), 6-3.
No. 2 seed Mikhail Youzhny eased past Argentina's Leonardo Mayer 6-1, 6-1. The Russian, who won the tournament in 2002, next plays the eighth-seeded Argentine Federico Delbonis, who defeated Germany's Benjamin Becker 6-7 (6), 6-4, 6-4.
The third-seeded Roberto Bautista Agut of Spain beat Louk Sorensen of Ireland 6-3, 1-6, 6-1, for a quarterfinal against compatriot Guillermo Garcia-Lopez. The No. 6 seed earlier defeated Swiss qualifier Yann Marti 7-6 (13), 7-5.
Home favorite Philipp Kohlschreiber won't be there. The German defeated compatriot Jan-Lennard Struff 6-4, 6-3 in their rain-postponed first-round match, but later lost 7-5, 7-6 (5) to Rosol in the second round.