Herbert's and Mahut's constant net rushing proved too much to handle for the Italian pair, which fell apart late in the opening set.
Fognini missed an easy overhead smash and Bolelli served consecutive double faults when Herbert and Mahut broke to take the first set.
The French pair won the 2015 U.S. Open and 2016 Wimbledon titles together while Fognini and Bolelli won the 2015 Australian Open.
Fognini replaced Paolo Lorenzi in Italy's doubles lineup.
In Sunday's reverse singles on the outdoor red-clay court, Fognini faces Lucas Pouille, and Andreas Seppi meets Jeremy Chardy.
"Hopefully the boys can get the job done in the singles," Herbert said. "It's going to be a great match with Lucas and Fognini."
Defending champion France is aiming for a spot in the last four for the third straight year.
WTA Charleston (4/2-4/8)
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Davis Cup Quarterfinals (4/6-4/8)
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