Fognini reached the fourth round of the recent Australian Open and then won both his matches in Italy's first-round Davis Cup triumph over Argentina.
He drew a first-round bye in this tournament and then advanced Thursday by defeating Aljaz Bedene 6-4, 7-6 (8-6).
Fognini will take a 6-2 record for 2014 into his quarterfinal match against Jeremy Chardy, the tournament's fifth seed who got past Carreno Busta 6-3, 7-6 (7-5).
In other second-round matches, third-seeded Nicolas Almagro was a 4-6, 6-3, 6-4 winner over Martin Alund, qualifier Taro Daniel moved on in his ATP Tour debut by upsetting eighth-seeded Federico Delbonis 1-6, 7-6 (7-5, 7-6 (9-7) and Daniel Gimeno-Traver ousted defending champion Horacio Zeballos 6-2, 6-4.
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