Fleury blanked the Blue Jackets until 56 seconds were left in the regulation, when Columbus added a sixth attacker and finally dented him on a score by Matt Calvert.
Seconds earlier, he just missed scoring an empty-net goal with a rink-length shot.
Fleury improved to 11-2-0 at home as the Penguins notched a sixth win in seven games.
Curtis McElhinney stopped 28-of-30 shots for Columbus.
Notable deaths of 2014 [PHOTOS]