"It's probably the biggest win of my career," said Dillier, whose best result this season was second place in the Dwars Door West-Vlaanderen one-day race.
Luxembourg's Bob Jungels retained the overall leader's pink jersey after finishing the 217-km ride from Reggio Calabria to Terme Luigiane in the main pack.
Adam Yates (Orica-Scott) is second six seconds behind with fellow Briton Geraint Thomas (Team Sky) in third, 10 seconds off the pace.
Friday's seventh stage will take the peloton over 224km from Castrovillari to Alberobello.
(Writing by Julien Pretot; Editing by Ed Osmond)
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