The Czech, seeded 19, downed the 25th seeded Italian 6-1, 6-4, 6-4 in just over two hours on Margaret Court Arena and will face either Roger Federer or Hungary's Marton Fucsovics in the last eight.
It will be 32-year-old Berdych's seventh quarter-final in his 15th Australian Open appearance and opens up the possibility of a 26th encounter with Federer.
The Swiss has won 19 of their 25 encounters. "What to say?" quipped Berdych who lost to Federer in the semi-finals at Wimbledon. "Let's see who's going to win. I have no choice to choose the opponent and may the better man win.
"I am going to get myself ready, it's all I can do and prepare the best I can. So far I think I've done pretty well. I'm feeling good, healthy, so I feel I am on the best base line I can be."
Berdych was in little trouble against the maverick Fognini, breaking the Italian's serve six times, hitting 37 winners.
"He's a very tough opponent," Berdych added. "You especially have to stay very focused from the first point to the last because you never know what's going to come up.
"He can always come up from the back of the court with amazing shots and turn the match around so being focused was the best thing I could have done today."
Berdych has reached the semi-finals on two occasions. Stan Wawrinka beat him on his way to the 2014 title. Andy Murray beat him in 2015.