The best-of-7 series is tied 3-3. Game 7 will be in Edmonton on Saturday.
"We gave it back to them a little bit, but we didn't break and we found a way to get one and I think it shows a lot of character in that group," Barrie said. "And now we've got home ice and we're headed back for Game 7. I think the boys are pretty pumped."
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Evander Kane scored twice and had an assist, and Connor McDavid had a goal and two assists for the Oilers, the No. 2 seed from the Pacific Division. Mike Smith made 30 saves.
"I thought our players really dug in and found a way to win," Edmonton coach Jay Woodcroft said.
Sean Durzi and Carl Grundstrom scored for the Kings, the No. 3 seed from the Pacific. Jonathan Quick made 33 saves.
"Somebody a few years ago said that usually the fourth game is the toughest one to win. Pretty sure you guys know who it is," said Kings captain Anze Kopitar, referring to current Calgary Flames coach Darryl Sutter, who led Los Angeles to the Stanley Cup in 2012 and 2014. "Obviously there's something to it. We would have liked to close it out, but it didn't happen, and we got to move on and look ahead to Saturday."
Barrie gave the Oilers a 3-2 lead with a wrist shot from the right face-off circle.
"(For) whatever reason, there was a big gap between me and [Leon Draisaitl], and I just saw that I could jump in and kind of make it an odd-man (rush), and he made a nice little sauce over to me and kind of got lucky to beat 'Quickie' there," Barrie said. … "So, just kind of a nice to see that one go in."