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Connor McDavid contributed a pair of assists for the Oilers, who evened the best-of-seven series at 1-1.
Smith, who made the fatal puck-handling mistake that led to Edmonton's 4-3 loss in Game 1 on Monday, collected the fifth playoff shutout of his career.
Jonathan Quick stopped 30-of-36 shots for the Kings.
The series now moves to L.A. for Game 3 on Friday, with the Kings also hosting Game 4 on Sunday. Game 5 goes Tuesday in Edmonton.
The Oilers put away their second power-play goal of the night 11:55 into the third after Quinton Byfield was called for holding.
Kane fired in a snap shot from inside the faceoff circle for his second goal of the night.
Edmonton jumped out to a 5-0 lead 3:23 into the third when Puljujarvi collected a pass across the slot from McDavid and ripped a shot over Quick's glove.
Just 21 seconds earlier, Kane capitalized when Sean Druzi bobbled a puck in the slot in front of the Kings' net and the Oilers winger tipped it in for his first goal of the post-season.
The Oilers took a 3-0 cushion into the third period after Evan Bouchard sent a bomb sailing from just inside the blue line and tipped it past Quick 16:05 into the second.
L.A. got their third man advantage of the night earlier in the period after Kailer Yamamoto was called for slashing, but it was Edmonton that found the back of the net.
McDavid and Nurse charged down the ice and Nurse fired a wrist shot that deflected off the Kings' Arthur Kaliyev and over Quick's glove to give the Oilers a 2-0 lead 6:03 into the period.
After a goalless first, Edmonton opened the scoring on the power play after Dustin Brown was called for goaltender interference early in the second.
Tyson Barrie sliced a crisp pass to Draisaitl in the far faceoff circle and the star sniper blasted a one-timer past Quick for his second goal of the playoffs.
The tally was Edmonton's third power-play goal of the series.
The Oilers were 2 for 3 with the man advantage on Wednesday and the Kings went 0 for 4.
L.A. controlled much of the action through the first, outshooting the home side 12-6 with five minutes left in the opening frame, thanks in large part to a pair of power plays.
Alex Iafallo had the best chance for the Kings 12:17 into the game with a backhander that ricocheted off the post before Smith kicked it out of the crease.
NOTES: L.A. is 0 for 8 on the power play through the first two games of the series … Kings winger Viktor Arvidsson missed a second straight game with an undisclosed injury … Winger Josh Archibald slotted into Edmonton's lineup as head coach Jay Woodcroft opted to play 12 forwards and six defencemen.
This report by The Canadian Press was first published May 4, 2022.
Gemma Karstens-Smith, The Canadian Press