Draisaitl's shootout tally came in the fourth round after Jussi Jokinen scored on the other end to keep the Canucks alive momentarily. Alexander Edler missed his shootout attempt to seal the Oilers' victory.
Draisaitl (25th goal) and Ryan Nugent-Hopkins (24) accounted for Edmonton's regulation goals and goaltender Cam Talbot made 36 saves as the Oilers ended their season on a two-game winning streak following a five-game skid.
Jokinen (5) and Tyler Motte (5) had a goal apiece and Anders Nilsson had 31 saves for Vancouver, which won five of its final seven games.
Brothers Henrik and Daniel Sedin each played their final NHL game after 17 seasons apiece with Vancouver. Daniel scored in the opening round of the shootout, but Henrik's attempt in the second round was off target.
The iconic brothers combined for 168 career points against the Oilers, the most combined points by any teammates against Edmonton in NHL history, according to Elias Sports Bureau.
Edmonton advanced to the second round of the playoffs last season, but failed the qualify for postseason play for the 11th time in 12 seasons dating back to the franchise's most-recent Stanley Cup Finals appearance in 2005-06.
Vancouver also fell short in pursuit of a playoff berth for the fourth time in five years. In three postseason appearances since their run to the Cup Final in 2010-11, the Canucks were eliminated in the first round each time -- most recently in 2014-15.
Draisaitl's power-play goal on a wrister from the left circle with 6:08 left in the first period put Edmonton ahead 1-0. Jokinen tied it for Vancouver with 2:13 to go in the period when he deflected Brendan Leipsic's shot past Talbot.
Motte fired in a rebound from the left circle to give the Oilers a 2-1 edge with 8:15 to go in the second. Nugent-Hopkins tied it for the Canucks with 5:50 before the second intermission on a wrister from the right circle off a nifty drop pass from Connor McDavid.
Both Sedins had scoring chances in overtime, with Henrik sending a puck off the post late.
The Oilers split their four-game season series with the Canucks.
--Field Level Media
|Jussi Jokinen 1||Points||Ryan Nugent-Hopkins 2|
|Jussi Jokinen 1||Goals||Ryan Nugent-Hopkins 1|
|Darren Archibald 1||Assists||Connor McDavid 2|
|N/A||Power Play Goals||Leon Draisaitl 1|
|N/A||Short Handed Goals||N/A|
|Anders Nilsson .939||Save Percentage||Cam Talbot .947|
|Anders Nilsson 31||Saves||Cam Talbot 36|