Stolarz, making his first start since Jan. 27, made three saves in overtime during a power play and then turned away Couture's attempt in the first round of the shootout.
Rakell won it in the third round, slowing up to the side of Reimer and backhanding the puck high in the net.
"I don't think I've done that [shootout move] before," Rakell said. "I've done it a lot in practice. It wasn't really what I had in mind."
Derek Grant gave the Ducks a 1-0 lead at 3:38 of the first period when he finished off a 2-on-0 rush with Sam Carrick.
Couture tied it 1-1 at 7:10. He won a face-off in the Anaheim zone and the puck went to Burns at the top of the left circle. Burns' shot was blocked by Ducks forward Ryan Getzlaf, but Stolarz lost sight of the puck and Couture was in position to push it into the net.
Burns made it 2-1 at 17:22 on a slap shot through traffic. It was his fifth goal of the season and first in 17 games.
Stolarz made back-to-back difficult saves on Timo Meier at 5:51 and 6:09 of the second before Rakell tied it 2-2 at 7:00.
Zegras took a shot off a face-off win by Adam Henrique. Reimer made the save, but the rebound went off Rakell and into the net.
"You've got to finish," Boughner said. "We had plenty of chances to finish and we didn't, whether it's a breakaway, whether it's a 2 on 1, whether it's a back door, 3 on 1. You've got to finish on some of those."