Pittsburgh closed the series in five games, with the Blue Jackets' only win coming in Game 4.
The Penguins built a 3-0 lead through the early part of the second period on a goal by Phil Kessel and two from Bryan Rust. Columbus responded with goals from William Karlsson and Boone Jenner to pull to within 3-2 before the end of the second.
Sidney Crosby scored on a one-timer from Evgeni Malkin for a power-play goal and some breathing room at 5:31 of the third before Scott Wilson scored 51 seconds later on a backward tip to give the Penguins an insurmountable 5-2 lead.
Goaltender Marc-Andre Fleury, who stepped in when Matt Murray got hurt in the warmup before Game 1, made 49 saves and became the all-time Pittsburgh leader with 57 career playoff wins.
Sergei Bobrovsky stopped 27 of 32 Pittsburgh shots.
Pittsburgh advances to play the winner of the Capitals-Maple Leafs series, which is tied 2-2 heading into Game 5 on Friday in Washington.
As much as the Penguins talked about getting a better start, Columbus again dominated the early minutes as it has in each game of the series.
The Blue Jackets had a 6-1 and 8-3 advantage in shots in the first five minutes, but Fleury thwarted those attempts.
Pittsburgh then began to take over the momentum and grabbed a 1-0 lead on Kessel's power-play goal at 9:07 of the first. From the top of the left circle, he found a sliver between Bobrovsky and the near post for a 1-0 lead.
Rust increased that to 2-0 at 1:07 of the second with a lofted, off-balance backhand from in close off of a Kessel rebound. Malkin picked up his second assist to extend his playoffs-leading point total to 10.
Rust struck again -- another backhander off of a rebound -- for a 3-0 Pittsburgh lead at 3:50 of the second.
Karlsson cut the Blue Jackets' deficit to 3-1 when his shot bounced in off of Fleury's right shoulder at 9:30 of the second.
Jenner further cut the lead to 3-2 with a power-play goal at 12:24 of the second. He knocked a rebound of a shot by Seth Jones out of the air and behind Fleury.
NOTES: Blue Jackets LW Nick Foligno missed his first game of the series because of a lower-body injury. ... Columbus D Zach Werenski was named one of three finalists for the Calder Trophy, which goes to the NHL's top rookie. He missed the last two games of the series due to a broken cheekbone. ... Penguins RW Phil Kessel and D Trevor Daley, who did not practice Wednesday, were in the lineup. ... Blue Jackets veteran LW Scott Hartnell took the lineup spot vacated by Foligno after being a healthy scratch for Game 4 in Columbus.