Goodrow, who signed with the Sharks (2-1-1) as an unrestricted free agent in March, has three goals in two games.
Sharks goalkeeper Alex Stalock stopped 16 shots, while John Gibson had 30 saves for the Ducks.
Couture opened the scoring midway through the first period, scoring on a power play off a pass from Jason Demers. Goodrow scored two minutes later to make it 2-0.
Chris Tierney set up Goodrow's second goal, also on a power play late in the second period, backhanding a pass to a wide-open Goodrow at the top of the crease.
Matt Beleskey scored midway through the third period for the Ducks (2-2-1).
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