Pittsburgh won its third in a row and reached 10 wins in its 14 games this season.
A power-play goal in the first period by Chris Kunitz put the Penguins in front and they then broke the game open with three goals during the first 5:34 of the second.
After Sutter snapped a shot by Sergei Bobrovsky, Kris Letang and Jayson Megna added goals for a 4-0 lead. Bobrovsky was removed after Letang's goal and was replaced by Curtis McElhinney, who allowed a goal on 14 shots.
Derek MacKenzie and Ryan Johansen produced the goals for Columbus, which will host Pittsburgh Saturday.
Marc-Andre Fleury picked up the win by stopping 37-of-39 shots.