Sawchenko had previously made four relief appearances this season, including in the loss to the Predators.
"Throughout my journey, that's the one thing I've really held close to me is that belief that I know I can play here," Sawchenko said. "Earning the trust to get a start, that's a whole another step, right, so it's all part of the process here, just brick by brick, building a career and trusting the process."
Couture gave the Sharks a 1-0 lead at 7:47 of the first period when he shot five-hole on Stolarz from the left circle on a 2-on-1.
Ducks forward Isac Lundestrom sustained a lower-body injury in the neutral zone prior to the goal and did not return. Anaheim was also without center Ryan Getzlaf, who missed the game because of a lower-body injury and is day to day.
"Those are two influences on our games, right, [Getzlaf] and [Gibson]," Ducks coach Dallas Eakins said. "Two big leaders in our room, and then losing 'Lundy' early in the game, everyone had to pick up."
Henrique, who moved into Lundestrom's spot at center, tied it 1-1 at 1:41 of the second period. Max Comtois' initial shot was saved by Sawchenko, but the rebound deflected in off Henrique near the left post.
"We had three power plays in the first period that we didn't click on," Henrique said. "They can get momentum off of that ... but I think we did a good job sticking with the game plan and kept pushing and pushing and pushing and trying to wear them down over the full game."