Twice champion Vincenzo Nibali (Bahrain-Merida) limited the damage as he finished one minute five seconds behind after a strong showing in the finale, a 12.25-km climb in San Marino, to take fourth place. Belgian Victor Campenaerts and Dutchman Bauke Mollema finished second and third respectively.
Briton Simon Yates (Mitchelton Scott) took a beating as he lost 3:11 to Roglic, and Spain's Miguel Angel Lopez ended up 3:45 off the pace after all the general classification contenders rode the stage in wet conditions.
"It's a perfect performance in my mind. I did a good job. I took it easy at the beginning and I gave it all at the end. It's nice to take some time over the other GC favorites but the Giro is far from over," said Roglic.
It was a spectacular reverse of fortune for Yates, who had been strong in the opening time trial after warning his rivals they should be scared of him.
Overall, Conti, who is not among the main GC contenders, leads Roglic, winner of the first time trial, by 1:50.
Roglic has a one minute 44 second advantage over Nibali in the overall standings before the Giro hits the mountains, and 3:46 on Yates.
The first mountain stage will be contested on Friday, which could reshuffle the overall standings as Yates, Nibali and Lopez are expected to attack during the most gruelling part of the race.
Monday is the three-week race's first rest day.
(Reporting by Julien Pretot; Editing by Pritha Sarkar)
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