Joe Thornton had a goal and an assist while Torrey Mitchell also scored for the Sharks (34-25-9), who are trying to get back into the playoff picture after losing 11 of their previous 13 games.
San Jose is one point back of eighth-place Colorado in the tight Western Conference race. The Avs won 3-2 in overtime against Anaheim on Monday.
Ryan Whitney and Taylor Hall scored for the Oilers (26-36-7), who have lost six of seven.
San Jose got off to a fast start, taking advantage when Oilers goalie Devan Dubnyk lost sight of a bouncing puck. Mitchell converted a rebound of a Tommy Wingels shot just 1:07 into the game.
The Oilers tied it with 49 seconds left in the first. Whitney wound up from the top of the faceoff circle and fired a shot past Antti Niemi while both teams were down to four players a side.
Dubnyk made up for his earlier gaffe midway through the second, coming across to make a huge stick save on a rebound attempt by T.J. Galiardi.
The Sharks went up 2-1 on the power play a minute later. Patrick Marleau fed a pass through to Thornton, who snapped home his 15th of the season.
Edmonton drew even again with five minutes to play in the second on a tremendous effort by Hall. After blocking a shot at his own blue line, Hall turned on the jets to outrace a pair of San Jose defenders to the puck and beat Niemi for his 26th of the season.
The Sharks got their go-ahead goal 38 seconds into the third period when Dan Boyle unleashed a hard shot from the point that Couture tipped past Dubnyk while on the power play.
NOTES: It was the fourth and final scheduled meeting between the teams this season. Edmonton won two of the first three in shootouts. ... San Jose had gone six games without a power-play goal before scoring with the man advantage in the second period. ... Oilers F Ben Eager returned to the lineup after missing two games with the flu. ... Sharks winger Marty Havlat could return as early as next week. He has been out since Dec. 21 with a hamstring injury. ... The last time the Sharks missed postseason play was in 2003.