Kapanen cut it to 2-1 at 6:55 when Evgeni Malkin's pass deflected in off his skate, and Carter tied it 2-2 at 7:04 when he scored from a sharp angle after intercepting a pass Jets goalie Connor Hellebuyck sent off the end boards.
"I think we had a lot of chances throughout the game," Kapanen said. "First two periods weren't our best. We still had chances. I think that [first goal] goes in for us, it's a big weight lifted off everybody's shoulders. … It was a good segment there."
Tristan Jarry made 27 saves for the Penguins (26-10-5), who have won five straight and are 16-2-0 in their past 18 games.
Blake Wheeler had a goal and an assist, and Hellebuyck made 36 saves for the Jets (17-14-7), who have lost four straight (0-2-2). Hellebuyck made 29 saves in a 3-2 loss at the Boston Bruins on Saturday.
"Game goes from a shutout, and what should be a shutout, to an overtime, shootout loss," Hellebuyck said. "The second [Penguins goal], I just had a late decision because it looked like it was going to be icing for a second. Then when I finally decided, my only play was to rip it behind the net and they just have a guy wide open right there. Just seems like my luck right now."