The BMC rider outsprinted breakaway companion Jasper Stuyven (Trek-Segafredo) in the final short climb, with Lukas Poestlberger (Bora-Hansgrohe) taking third place 12 seconds back.
"Basically the stage started pretty bad for me. I had a flat tire at KM 0. I chased back and I could go straight away in the breakaway,” Dillier said.
“It was a really hard stage at over 200km. We had tail wind for most of the stage but it doesn’t mean it was easy because we always had to push hard to stay away in little climbs.
“It was for sure a tough day. To sprint at the end wasn’t easy as well. On the profile, it didn’t look like as stage for the sprinters. The last 1.5km uphill wasn’t for pure sprinters, but there aren’t many Classics specialists in the peloton, that’s why I took my chance.
“I have a powerful sprint. I know this kind of finish suits me well. I just tried to believe in myself. Jasper [Stuyven] is a really strong rider. To beat him is a nice feeling.”
Bob Jungels (Quick-Step Floors) retained the overall leader's pink jersey after finishing the 217km stage from Reggio Calabria to Terme Luigiane in the main bunch.
Adam Yates, riding for the Australian Orica-Scott team, is second six seconds behind with Geraint Thomas (Team Sky) in third, 10 seconds off the pace.
Dillier, Stuyven, Pöstlberger were the remains of an original five rider break to make the tough 1.5km uphill finish to Terme Luigiane.
The quintet of riders crafted an advantage of more than eight minutes as they tackled the first categorised climb of the day.
By the time the race entered the final 100km the advantage had reduced to less than six minutes but continued their resistance on a course which favoured a breakaway winner.
Approaching the final 25km the five still had a three-minute advantage but as they hit the final categorised Fuscaldo climb, lost Mads Pedersen (Trek-Segafredo) and Simone Andreetta (Bardiani CSF) to the pressure.
Stuyven attacked over the summit but Dillier and Pöstlberger were able to stay on his wheel as they headed toward the finish.
With the peloton closing in fast, the trio dug deeper for the final kilometre while jockeying for position.
It was a stalemate until the final 250 metres when Dillier launched his winning move to take the biggest win of his career.
Friday's seventh stage will take the peloton over 224km from Castrovillari to Alberobello.