The Penguins trailed 3-0 after two periods but came back to tie it before Picard scored with 2:02 to play for the Blue Jackets. But Kunitz answered 17 seconds later to send the game to overtime.
Less than a minute into overtime, Kunitz took a pass in the slot from Crosby, who had stolen the puck behind the net.
Mayorov scored twice in a span of 4:24 late in the second period.
The game was originally scheduled for Sept. 25 but moved back because Pittsburgh is hosting the G-20 summit of world leaders that day and traffic is heavily restricted around the city.
The Penguins just reported to camp Saturday -- two days after being honored at the White House by President Barack Obama. They went through only their third practice of training camp earlier in the day Tuesday.