It is the fastest goal to open a period in Hurricanes/Hartford Whalers history.
Martinook got to a loose puck off the opening face-off, split Flyers defensemen Travis Sanheim and Ivan Provorov, and beat Hart with a shot from the left circle.
[WATCH: All Flyers vs. Hurricanes highlights]
"I took step up and felt I could get by them," Martinook said. "Put a shot on net. Think it went off his glove, but I'll take it."
Lucas Wallmark and Andrei Svechnikov scored for Carolina (16-17-5), which won for the first time in three games. Curtis McElhinney made 22 saves.
Jakub Voracek scored for Philadelphia (15-18-5), which has lost three games in a row. Michal Neuvirth made 23 saves after replacing Carter Hart 2:19 into the second period. Hart allowed three goals on 10 shots.
Video: PHI@CAR: Svechnikov snaps home wrister from the slot
The Flyers started the second period with four seconds of power-play time, and Hurricanes coach Rod Brind'Amour said he felt his team's aggressive approach on the face-off caught Philadelphia by surprise.
"That's the goal of the game," Brind'Amour said. "Start a period with four seconds on the power play, they're not thinking that's going to happen. Maybe caught them sleeping a little bit. That was huge."
The goal was the ninth of the season for Martinook, who has produced playing on all four lines. He played Monday at left wing with Wallmark and right wing Justin Williams.
"I do the same thing no matter what line I'm on," he said. "Not going to try to play like [Wayne] Gretzky when I'm with [Sebastian] Aho and [Teuvo Teravainen] or anything. I do what I do, and usually it's good enough for me."
Video: PHI@CAR: Wallmark wrists one by Hart
Wallmark gave Carolina a 1-0 lead at 10:47 of the first period with a shot from above the face-off circles.
It was the first time in six games the Hurricanes scored the first goal.
"We've been struggling to score," Martinook said. "Sometimes you need that ... not sure it's a good shot, it's from far out and sometimes you need one to get you jump-started, and hopefully that's the one that can get us going."
After Martinook scored, Svechnikov made it 3-0 at 2:19 of the second with a shot from between the top of the circles. It was his 11th goal, second among NHL rookies after Vancouver Canucks forward Elias Pettersson's 19.
Hart, a rookie making his fifth start since being recalled from Lehigh Valley of the American Hockey League on Dec. 17, has lost three straight after winning his first two NHL games.
"Just three pucks that I should have," he said. "Going forward, I'll learn from it. Put it behind me and move on to the next one, that's the name of the game. There's going to be nights that don't go our way; tonight was one of those nights."
Video: PHI@CAR: Voracek, Simmonds team up for smooth tally
The Flyers are 3-3-1 since coach Scott Gordon was hired Dec. 17 to replace Dave Hakstol. He said they played a good first period but got away from what had worked in the second.
"I thought our first period was one of the best periods we've had," Gordon said. "Didn't give up many shots, didn't give up many scoring chances. Thought our transition was great coming out of our zone. And then it changed. We stopped advancing the puck, started playing behind the goal line, and that was one of the things when I got here, we talked about trying to get out of our zone as quick as we can. The first period we did it and after that a little better in third. But in the second period, that's what cost us the game."
Voracek got Philadelphia to within 3-1 at 2:38 of the third period. He has a point in five of the past six games (two goals, three assists).
They said it
"It's tough to bounce back in this league. Got to be better, everybody's got to be better tomorrow because it's one of the top teams (Nashville Predators). Have to give better effort than tonight." -- Flyers goalie Michal Neuvirth
Video: PHI@CAR: Neuvirth denies McGinn with tough pad stop
"The game didn't feel much different than any of the games we've played all year. Just a couple went in and we were ahead, so it was a different game. We've had good goaltending all year, and same tonight. We just got a couple goals." -- Hurricanes coach Rod Brind'Amour
Need to know
Flyers forward Wayne Simmonds played his 800th NHL game. ... Ron Francis held the Hurricanes/Whalers record for fastest goal to open a period, scoring for Hartford eight seconds into the third against the Edmonton Oilers on Feb. 12, 1984. ... McElhinney played his 200th NHL game. ... Williams played his 1,200th NHL game. He's the fifth active player (Patrick Marleau, 1,614; Joe Thornton, 1,525; Matt Cullen, 1,473; and Zdeno Chara, 1,443) and 110th in NHL history to play at least that many.
Flyers: At the Nashville Predators on Tuesday (8:30 p.m. ET; NHLN, SNE, SNW, SNP, FS-TN, NBCSP, NHL.TV)
Hurricanes: At the Philadelphia Flyers on Thursday (7 p.m. ET; NBCSP, FS-SE, NHL.TV)
Video: Balanced offense propels Hurricanes to 3-1 win