Trailing 3-2, Arizona had three consecutive power plays to start the third period but couldn't score against the top penalty-killing unit in the NHL.
"[The penalty kill] has won us so many games," Couture said. "So credit to the penalty killers; the system's been great and our goaltenders have been spectacular."
Arizona was held to two shots on its three power plays.
"They're the No. 1 penalty killer and it's for a reason; they're good," Coyotes center Derek Stepan said. "But I keep going back to the word execution. Down a goal there in the third period, we have to execute."
Meier gave the Sharks a 3-2 lead at 8:52 of second period when he redirected Marc-Edouard Vlasic's slap shot from the left point following Tomas Hertl's face-off win.
Gambrell tied it 2-2, switching from his backhand to his forehand while skating from behind the net for a wrist shot under the crossbar at 2:55 of the second.
Couture started the Sharks' comeback at 11:01 of the first with a wrist shot from the right circle that made it 2-1.
"The way we started, it could have been 4-0, 5-0, but I thought we settled down and got to our game," Sharks coach Peter DeBoer said. "I think the next 50 minutes were probably some of the best hockey we've played this year."
The Coyotes scored on their first two shots. Schmaltz finished off a tic-tac-toe passing sequence to score from the edge of the crease for a 1-0 lead at 1:33, and Stepan made it 2-0 with a goal 39 seconds later at 2:12.
"We had a breakaway (by Michael Grabner) to go up 3-0, and after that it just kind of unraveled. You could tell guys were tight, and I'm not sure why," Tocchet said.
Couture scored into an empty net with 29 seconds remaining for the 4-2 final.
San Jose was coming off a 4-1 win against the Los Angeles Kings on Friday.
"Beating L.A. [Friday] night at home and then coming in here and beating [the Coyotes], who's above us in the standings, these are opportunities to close the gap," DeBoer said. "We wanted to take advantage of both those games."
