The Italians, however, rebounded in front of their home fans as Fognini charged back following a poor first set to overcome Dodig 4-6, 6-4, 6-4, 6-4.
Fognini logged six service breaks in the 2-hour, 52-minute match.
Italy now moves to the April quarterfinals. There they will meet Canada, which ousted Spain Sunday in another first-round match.
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