Couture also scored twice in regulation, helping the Sharks snap a two-game skid. Couture's goal 6:50 into the third period cut San Jose's deficit to 3-2 and Tommy Wingles tied it midway through the stanza.
That was all the scoring until the shootout, in which Couture and Edmonton's Jordan Eberle traded goals in the second round. Boyle then beat Devan Dubnyk with the deciding shot.
Couture ran his goal total for the season to 14.
Antti Niemi made 24 saves to get the win while Dubnyk was touched for three goals on 34 shots.
San Jose has won on the road just five times in 16 tries this year.