The Blue Jackets built a 3-1 advantage under a minute into the second period, but Pittsburgh, which finished tied for the NHL's top power-play unit in the regular season, netted a pair of power-play goals 45 seconds apart to tie the game less than 2 1/2 minutes into the middle stanza.
Sutter supplied the winner with a wrister from the right circle, while Marc- Andre Fleury made nine of his 31 saves over the final 20 minutes to help the Pens draw first blood in this best-of-seven set.
Beau Bennett and Matt Niskanen each notched a goal and an assist, while Jussi Jokinen also lit the lamp for the Metropolitan Division champion Penguins, who went 0-5-0 against Columbus during the regular season.
Jack Johnson collected a goal and an assist, while Derek MacKenzie and Mark Letestu tallied a goal apiece for the Blue Jackets, who qualified for the postseason for just the second time in franchise history, with the other appearance ending in a four-game sweep at the hands of Detroit in 2009.
Sergei Bobrovsky stopped 28 shots in the setback.
Game 2 of this series is scheduled for Saturday in Pittsburgh.