Two early breaks in the first set allowed Fognini to take control at 1-5, but Beck secured a break of his own in the middle of two holds to leave the score on serve at 5-4.
Fogini held his nerve, however, and served the first set out in style.
It was a similar story in the second set, with Fognini securing two breaks to Beck's one. As a result, the Italian secured a two-set lead with another 6-4 score.
The third and final set proved to be more straightforward, however, with three breaks of serve and three holds allowing Fognini to advance into the next round.