Jiri Tlusty and Elias Lindholm also scored for the Hurricanes, who lost the first four games since the Olympics before capping their five-game road trip with a win. Carolina also lost its final game before the break and has fallen to 13th place in the Eastern Conference — six points out of the final playoff spot.
Anton Khudobin made 35 saves to earn the win just hours after signing a two-year, $4.5 million contract extension.
Marty Havlat and Brent Burns scored for the Sharks, who lost in their first home game since Feb. 7. Alex Stalock made 32 saves in his first back-to-back starts of the season.
Both teams had chances early in overtime before the Hurricanes capitalized in the final minute on a scramble in front of Stalock. Lindholm took possession of a blocked shot and slid a backhand pass through the crease to Harrison, who knocked in the winner for Carolina's fourth straight victory in this series.
The Hurricanes tied the game early in the second period when Eric Staal found Lindholm alone in the slot with a pass from behind the net for a one-timer.
The Sharks kept the puck in Carolina's defensive zone for much of the second half of the period but couldn't generate many quality scoring chances, and the game remained tied heading into the third.
Carolina struck first when Tlusty took a pass from Alexander Semin, split defenseman Jason Demers and Scott Hannan and skated in alone on Stalock before beating him with a backhand shot.
Scoring first has been a good omen this season for the Hurricanes, who have won more than three-quarters of their games after getting the opening goal. But this lead was short-lived when Khudobin allowed a big rebound on a shot by Tyler Kennedy. Havlat knocked it in while falling to the ice for the equalizer.
The Sharks took advantage of another misplay from Khudobin to take the lead with 28 seconds left in the period. James Sheppard took the puck away from the goalie behind the net and sent a backhand pass in the slot to Raffi Torres, who teed up Burns for his first goal in 20 games to make it 2-1.
NOTES: Carolina D Andrej Sekera had two assists and has points in all five games since the Olympic break. ... The Hurricanes were without D Justin Faulk, who suffered an upper-body injury on Sunday in Anaheim. ... The Sharks honored their four Olympians with a video board tribute in the first period.