Couture's winner gave the Sharks their second victory of the preseason just 32 seconds after the Coyotes were called for having too many men on the ice.
Oliver Ekman-Larsson gave the Coyotes a 1-0 lead with 9:52 left in the first when he got a wrist shot past Harri Sateri. Just 52 seconds into the second, the Sharks capitalized on a four-minute double-minor penalty to Chris Brown and tied it up when Freddie Hamilton quickly scored on a power play.
Jason Demers gave San Jose a 2-1 lead when he scored on a power play with 8:23 remaining in the second as he was able to skip the puck past Mike Smith. Phoenix evened the score with 31 seconds left in the second when Andy Miele converted a power play.
Sateri is trying to win a spot on San Jose's roster as the backup to regular goalie Antti Niemi. He made 24 saves and Smith had 23.