Draisaitl and Slepyshev each buried their chances for their first career playoff goals and the Oilers held off the Sharks after that to get the win in their first playoff series since getting to the Stanley Cup Final in 2006. Connor McDavid iced the game with an empty-net goal with less than a second to play for his first even-strength point of the series.
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Patrick Marleau got the Sharks on the board with 7:48 left in the third period, but Cam Talbot didn't allow anything else on 28 shots to get the win.
Edmonton will take a young roster led by NHL scoring leader McDavid and eight other skaters under the age of 25 into the second round against Anaheim.
The defending Western Conference champion Sharks had a quick exit this year in the playoffs as they never fully recovered after blowing a two-goal lead in Game 5 in Edmonton.
Martin Jones made 18 saves, but couldn't stop the two breakaways and the offense failed to deliver in what could have been the final home game for longtime Sharks greats Joe Thornton and Marleau. The 37-year-old forwards are both free agents this summer and it is uncertain whether the Sharks will bring them back for another run.
The Sharks nearly got the equalizer on a late power play, but Joe Pavelski's backhand attempt hit the cross bar and post.
This game turned in the opening minutes of the second period. It started when Oscar Klefbom blocked Justin Braun's point shot and Adam Larsson sprung Draisaitl on a breakaway. Draisaitl held off Braun and then beat Jones between the pads to make it 1-0.
After having no points in the first four games and getting ejected in Game 4 for a spearing penalty, Draisaitl played a big role in the final two wins with a pair of assists, including one on the OT winner in Game 5, and the first goal in the clincher.
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Just 56 seconds later, Slepyshev got into the action when Paul Martin mishandled a pass from Chris Tierney at the point. Slepyshev raced to the puck and got in alone on Jones on the goal to make it 2-0 just 1:50 into the period.
The Sharks finally found their game after that and put on heavy pressure late in the second period, but Talbot made a nice save on Marleau in alone and Marcus Sorensen hit the cross bar on another chance.
After dominating the overtime in their Game 5 win, the Oilers picked right up from there and controlled the play in the opening period of Game 6. They had the decided edge in shot attempts and scoring chances but neither team was able to score.
San Jose's best chance came short-handed when Talbot stopped Melker Karlsson alone in front.
NOTES: Klefbom did not play the third period for Edmonton. ... F Joonas Donskoi returned to the lineup for the Sharks in place of Timo Meier.
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