Rick Nash netted his team-leading 24th goal of the season and added an assist while Steve Mason made 32 saves to contain the Penguins, who were playing without injured forwards Sidney Crosby and Evgeni Malkin.
Samuel Pahlsson, Jakub Voracek and Antoine Vermette also scored for Columbus while Tyler Kennedy scored the lone goal for the Penguins.
With the win, the Blue Jackets (26-22-5) moved to within four points of the Calgary Flames for the eighth and final playoff spot in the NHL's Western Conference.
Pittsburgh (34-17-4) has lost two straight games and trail the Eastern Conference-leading Philadelphia Flyers by three points.
Crosby was out of the lineup for the 14th straight game with a concussion while Malkin was lost last week to torn ligaments in his right knee.
(Writing by Jahmal Corner in Los Angeles; Editing by Frank Pingue in Toronto)