Pouille battled past the Italian number one 2-6, 6-1, 7-6 (7/3), 6-3 to give France an unassailable 3-1 lead in the quarter-final.
Sunday's redundant second reverse singles will not be played, organisers announced.
The title holders will face either Spain or Germany in the last-four tie in September.
Nicolas Mahut and Pierre-Hugues Herbert had cruised to a straight-sets victory over Fognini and Simone Bolelli in the doubles on Saturday to put France in control.
Pouille had clinched France's first point on Friday with a five-set success over Andreas Seppi with Fognini levelling in a bad-tempered four-set affair against Jeremy Chardy.