Kunitz was credited with the goal after the puck caromed off two Blue Jackets players on the power play late in the second period.
Nick Johnson scored from between the circles on a power play for the winning goal at 8:13 of the third, and Mike Comrie made it 5-3 four minutes later with a tap-in, finishing off a nice passing play.
Arron Asham and Mike Rupp also scored for Pittsburgh (2-0).
Tomas Kubalik had three goals, his third with 44 seconds left, and Kyle Wilson also scored for Columbus (1-2).
Pittsburgh's Brent Johnson allowed three goals on 21 shots, playing the first two periods. The Blue Jackets' Mathieu Garon went the distance and made 24 saves.
The teams will meet again Saturday night in Pittsburgh.