Sharp and Marian Hossa also scored 2:22 apart in the first period for the defending champions, who blew a 2-0 lead in the third before winning in the tiebreaker.
Corey Crawford finished with 32 saves, then stopped three Carolina shooters in the shootout.
After Patrick Kane and Jonathan Toews came up empty, Sharp beat Cam Ward with a nifty forehand in the third round.
Crawford then stuffed Jeff Skinner to end it, making the Blackhawks the last of the Original Six to reach the 2,500-win mark.
Alexander Semin and Ron Hainsey scored during Carolina’s third-period rally.
Ward stopped 34 shots for Carolina, which wrapped up a winless three-game homestand against Western Conference teams.
Chicago’s lead could have been even bigger had Ward not stopped a pair of Blackhawks from point-blank range just before time expired in the second.