The second-seeded Fognini won for the 20th time in his last 21 clay-court matches and advanced to a second-round tilt with Leonardo Mayer of Argentina.
Fognini also extended his current overall winning streak to seven matches. Fognini lost serve four times Tuesday, but broke Reister seven times in 13 opportunities.
Mayer moved on by ousting fellow Argentine Facundo Arguello 6-3, 6-4.
Sixth-seeded Robin Haase also advanced by outlasting another Argentine qualifier, Martin Alund, 6-4, 6-7 (6-8), 6-3.
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Michael Sam cut from the Dallas Cowboys