The world No. 10 broke serve on five occasions to win the match in 68 minutes.
"The more matches I play, the better I feel," Berrettini said. "The conditions are pretty slow here, so you need to make sure not be too passive like at the end of the second set.
"But besides that, I felt good. I played Bublik on the ATP Challenger Tour in Istanbul in 2017. He serves very well. He's unpredictable."
After winning a thrilling match which lasted almost three hours, Chardy will face Jan-Lennard Struff for a place in the semifinals.