"It's obviously very important for us," said Hedberg, who shut out the Washington Capitals on Monday. "We're in a situation where we need points. It was a tight game between two defensive teams. It felt in the third that this might go for a long time. It was a huge. A great shot by Kovy and a big win for us."
Kovalchuk, who didn't record a shot in regulation, beat Swedish rookie Henrik Karlsson with a wrist shot to the top corner on the first attempt for New Jersey (7-13-2).
It was somewhat of a surprise choice by rookie coach John MacLean. The last time he used Kovalchuk in a shootout, the $100 million man lost control of the puck on New Jersey's final attempt in a loss to the Buffalo Sabres.
MacLean didn't just go with Kovalchuk -- he was the leadoff man.
"No hesitation," MacLean said. "We prepared for it. We talked about it. I was very comfortable with him going. It was a great shot, and Heddy did his job on the other end."
Kovalchuk went straight at Karlsson instead of weaving his way toward the net, as he did the last time.
"I changed it up a little bit, so it worked better than last time," Kovalchuk said. "I really wanted to go. I can do better than I did last time."
David Clarkson netted the Devils' only goal in regulation as New Jersey spoiled the return of former coach Brent Sutter, who is now in charge of the Flames.
"It's all about two points," Kovalchuk said. "Both teams were desperate."
Rene Bourque scored for Calgary, which lost its fifth straight road game.
Mattias Tedenby lost control of the puck on the Devils' second chance, but Hedberg stood tall against Morrison on the Flames' final attempt.
Calgary is 2-8-2 in its last 12 games, including 1-1-2 in its last four.
"It's four games in a row now we've played well," said Sutter, who coached the Devils in the 2007-08 and 2008-09 seasons before taking the Calgary job. "We've been able to get four points out of a possible eight. We need to accumulate points. We know we need wins, but for that to happen you have to play some decent hockey, and we have. In the last two games we haven't scored enough goals but we've done other things well."
Sutter was pleased that his team limited New Jersey to 15 shots in regulation, but the Flames couldn't convert their scoring chances.
Karlsson made the save of the game in overtime, stopping a deflection by Patrik Elias in close on a great setup pass by Jason Arnott. The 26-year-old Karlsson had six saves in overtime, which included a Devils power play.
"He stepped up," Sutter said of Karlsson, who was making his third start. "He played well the other night. He hadn't played in a while, and I thought he responded well. He made a couple of big saves, and the one in overtime on Elias, it was a huge save. He had a strong solid game."
Hedberg nursed a 1-0 lead into the third period before Bourque snapped the goalie's shutout streak with his 10th goal of the season.
Defenseman Mark Giordano took a shot from the right point, and Bourque put the puck into the top part of the net with a deflection from at least 20 feet in front.
Karlsson, who gave No. 1 goalie Miikka Kiprusoff a rare night off, was also somewhat unlucky on the Devils' goal.
New Jersey defenseman Matt Corrente took a shot from the right point shortly after a penalty to Morrison ended. The shot appeared to be blocked and went to Tedenby. His shot from between the circles deflected off the skate of Clarkson and into the net.
The play was reviewed, and the call on the ice stood.
While neither goaltender was tested much, both made big saves.
Karlsson had a glove save on Elias early and a big stop on a power-play blast by Brian Rolston late in the second period.
Hedberg had a nice glove save on Tanguay late in the first period and good saves against Bourque, Tanguay and Jay Bouwmeester during a power play early in the second.