Berrettini, who recorded his ninth win in his last 10 matches, will take on Australia's John Millman, who defeated Dusan Lajovic of Serbia 6-3, 6-4, in the second round.
Basilashvili, winner of the Qatar ExxonMobil Open and BMW Open this year, got the decisive breakthrough enroute to clinching the first set with a forehand winner.
But Berrettini responded with two breaks and 10 winners in the second set. The 25-year-old went on to save three break points at 3-3 in the decider, and secured his fourth win over Basilashvili, when the world No. 30 mishit a forehand wide.
ALSO READ | Buffon to end long stay at Juventus
Italian Lorenzo Sonego came through another tough battle against No. 14 seed Gael Monfils of France, winning 6-4, 5-7, 6-4 in two hours and 48 minutes.
Sonego, who is now 12-8 this year, including singles and doubles titles at the Sardegna Open in Cagliari, struck 29 winners. He now challenges No. 10 seed Roberto Bautista Agut of Spain.
Bautista Agut posted a 6-3, 6-4 win over American qualifier Tommy Paul in one hour and 31 minutes. It was a repeat of their scoreline at the Rolex Monte-Carlo Masters last month.