Couture scored 11 seconds into the first period, the third-fastest goal to start a game in Sharks history, for a 1-0 lead.
Anaheim appeared to win the opening face-off, but San Jose forward Evander Kane poked the puck away from Ducks defenseman Jani Hakanpää at the Anaheim blue line and took it behind the net.
Kane's wraparound caromed into the slot, where San Jose defenseman Mario Ferraro shot into traffic. The puck got behind Miller and Couture poked it into the net before Hakanpaa could sweep it away.
Stephane Matteau scored for the Sharks 8 seconds into a 5-2 loss to the St. Louis Blues on Jan. 11, 2000, and Tyler Kennedy scored 9 seconds into a 4-3 shootout loss to the Dallas Stars on Oct. 17, 2013.
Lundestrom deflected a slap shot from defenseman Hampus Lindholm to tie it 1-1 at 5:42 of the second period.
"[First NHL goals] are always fun to see, and obviously the guys are excited on the bench for him," Eakins said. "That's a special time for that young man. I thought he was solid tonight. I thought him, Comtois and [Rickard] Rakell had lots of looks at the net and gave us some good O-zone time."
The Sharks, who have not played a home game this season, visit the Los Angeles Kings on Tuesday and Thursday before their home opener scheduled for Feb. 13 against the Vegas Golden Knights.
San Jose got at least one point in consecutive games for the first time.
"We still had a chance to take all four points out of here," Boughner said. "Taking the positives, we took three out of four, get a day off to rest and go to L.A."
NOTES:Sharks forward Patrick Marleau played his 1,733rd NHL game to tie Jaromir Jagr for third behind Gordie Howe (1,767) and Mark Messier (1,756). … Sharks forward Rudolfs Balcers, recalled from San Jose of the American Hockey League, had three hits and a blocked shot in 11:53 in his debut. ... Ferraro replaced Marc-Edouard Vlasic on the top defense pair with Brent Burns. ... Ducks forward Ryan Getzlaf missed his second straight game with a lower-body injury. He is day to day. … Ducks forward Sam Steel played his 100th NHL game. .. Miller is 388-282-85 with one tie in 18 seasons with the Ducks, Vancouver Canucks, St. Louis Blues and Buffalo Sabres.