The Canes played the third period without center Eric Staal, who didn't return after the second intermission because of a lower-body injury.
"This was definitely a gutsy effort and we really showed a lot of heart," said Leighton, who had 37 saves in picking up his first win since beating the Washington Capitals on Dec. 7. "All around, it was a great game for us. To come into the building of one of the league's best teams and win the game shows what this team is capable of.
"We worked hard for this one. We had some guys out of the lineup but we did everything we needed to do to be successful. We were jumping around after the game like we had won the Stanley Cup."
Tuomo Ruutu and Sergei Samsonov scored shootout goals for the Canes, with Samsonov clinching the road victory with a backhand shot past goalie Evgeni Nabokov. The Canes picked up perhaps their biggest road victory of the season two days after one of their toughest losses -- the 4-3 defeat in Vancouver on a late shorthanded goal by the Canucks.
"We played one of the best in the league," Samsonov said. "It was a good measuring stick for us and we hung in there."
The Canes got second-period goals from Matt Cullen, who scored on a penalty shot, and Samsonov. Forward Jakub Petruzalek, called up Thursday from the Albany River Rats and making his NHL debut, assisted on Samsonov's goal for his first NHL point.
When Ray Whitney scored at 7:30 of the third period — Cullen with the assist on Whitney's 17th goal — the Canes had forced a 3-3 tie.
But the Sharks took a 3-1 lead on Marc-Edouard Vlasic's power-play goal. Vlasic beat Leighton backdoor on a pass from Clowe at 7:16.
Some hustle in the San Jose zone by Petruzalek led to Samsonov's goal. Petruzalek raced to the boards to collect the puck and zipped a pass in front to Samsonov, who tipped in the puck for his first goal in 11 games.
With the Canes missing Brind'Amour and Scott Walker, Brandon Sutter centered the third line
with wingers Ryan Bayda and Chad LaRose. Ryan centered the fourth line with Patrick Eaves and Petruzalek.
The Canes hoped to have defenseman Niclas Wallin healthy enough to jump back into the lineup. But Wallin again was a scratch and Bryan Rodney again was needed.