Fognini will be eyeing the scalp of the out-of-sorts world number one in Rome following a straightforward 6-1 6-3 over wildcard Berrettini in his homeland on Sunday.
Guido Pella consigned Fognini to a first-round exit at the BMW Open last time out, but the world number 29 won 15 points in a row to open up a 4-0 lead in a flash at Foro Italico.
Fognini did not face a solitary break point in the match and broke for a third time to wrap up the first set before grasping the chance to serve for the match after Berrettini went 5-3 down having failed to hold.
Isner also reached round two after coming from a set down to beat Ramos-Vinolas 6-7 (4-7) 7-6 (7-4) 6-1.
Ramos-Vinolas exceeded expectations by reaching the final of the Monte-Carlo Masters last month, before he was beaten by Murray in the last eight of the Barcelona Open.
The Spaniard endured a setback on day one this time around as Isner produced a barrage of aces to keep his opponent at bay.
Fernando Verdasco and Jiri Vesely also progressed at the expense of Donald Young and Dan Evans respectively.
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