The 175.5km stage from Saint-Die-des-Vosges came down to a sprint finish as the quickmen kept pace with the peloton over the late climb of Cinq Chateaux.
Bevin is the first competitor to be forced out of this Tour.
The previous wearer of the iconic yellow jersey, French rider Julian Alaphilippe, did everything he could to keep it, battling up the ascent, through the dust.
Mitchelton-Scott's Trentin launched his sprint first but Sagan had the power to come around.
Great Britain's Geraint Thomas (R) rides with cyclists during the sixth stage of the 106th edition of the Tour de France cycling race between Mulhouse and La Planche des Belles Filles, on 11 July 2019.
Dan Martin and Nicholas Roche both finished in the leading group.
On stage 6 he showed his talents with a day-long break and second place behind Dylan Teuns (Bahrain Merida).
But an eight-second bonus for being first at top of the penultimate climb, the Col des Chevreres, and another six seconds for his second place meant that he moved past Alaphilippe as he ended up 1:35 behind at the top of La Planche des Belles Filles. Meanwhile, Ciccone goes down as the 269th yellow jersey in its 100-year history.
"I have to thank all my team-mates", Sagan said.
The seconds Thomas gained were vital given the question marks over his form and today's result means the former Tour de France victor jumped from seventh to fifth.
"When Movistar went and (Alejandro) Valverde were riding it was solid, I was feeling pretty good - I was just unsure as on those steep climbs I was expecting Richie Porte, Nairo Quintana and obviously Egan to jump up there, so I was hoping for it to be hardest all day before that".
Much had been made of the five seconds Bernal picked up due to a momentary lapse from Thomas in the finale of stage three in Epernay, and that gap was only expected to grow as gradients hit 24 per cent and the surface turned to gravel.
"It's one of those climbs where you really have to be patient", the 33-year-old Thomas said.
Friday's seventh stage is a 230km flat ride from Belfort to Chalons sur Saone.