The Stars played their most complete game of the Western Conference Final on Sunday, yet still saw their season end in a 2-1 loss to the Edmonton Oilers.

Dallas had a 35-10 advantage in shots on goal (including 14-2 in the third period), a 74-24 edge in shot attempts and more than three times the scoring chances of the Oilers, but a historic penalty kill and the strong work of goalie Stuart Skinner kept the puck out of the net just enough to close the series in six games.

The Oilers will now advance to the Stanley Cup Final and face the Florida Panthers.

“You could probably argue that was our best game of the series,” said captain Jamie Benn. “It just didn’t go our way.”

Jamie Benn speaks to the media after Game 6

Oilers goalie Stuart Skinner received a lot of accolades for his performance, stopping 34 shots, including 13 in the third period.

"We're on a plane to Dallas if it wasn't for Stu," Oilers captain Connor McDavid said of a potential Game 7 at American Airlines Center on Tuesday. Even Stars players felt they were close to tying the game.

“There wasn’t a moment until the buzzer just now where I thought we weren’t going to win or didn’t deserve to win,” Tyler Seguin said in the dressing room after the game. “I thought we could earn it, but we didn’t. That’s what playoff hockey is.”

That’s the hard part for the Stars. They battled to get past the two most recent Stanley Cup champions in Vegas and Colorado in the first two rounds. They bounced back from losing Game 1 against the Oilers and had a 2-1 lead in the series. They then took a 2-0 lead in Game 4 in Edmonton and seemed to have the Oilers and Skinner on the ropes. But they were outscored 10-2 the rest of the way by a scrappy Edmonton team that played great team defense and scored four power play goals in the final two games.

“You have to give Edmonton credit,” Stars coach Pete DeBoer said. “Their power play over the last two games was good, and their goaltender was good. It’s fine lines to get to this point of the year. They were on the right side of it. I thought we had a little bit of an off night in Game 5, and they won. They had a little bit of an off night tonight, and they still found a way to win. That’s the difference.”

Pete DeBoer on his "special group" after Game 6 loss

Edmonton scored on early power play goals by McDavid and Zach Hyman, and then held on for dear life in the final 44 minutes. The McDavid goal will be plastered all over the highlight reels as he weaved through three Stars players and beat Jake Oettinger with a backhand shot. Later in the period, McDavid set up Hyman for his 14th goal of the playoffs on a quick release that went top shelf.

Mason Marchment was able to punch home a rebound in the third period, but that was the only scoring for the Stars on the night.

Dallas ended up 0-for-14 on the man advantage in the series, and that was part of a record 28 consecutive kills for the Oilers.

It just added to the frustration, as the Stars had several opportunities to tie Game 6 – or one of the previous games – and just couldn’t cash in.

“They did a good job defending,” said center Wyatt Johnston, who led the Stars in the playoffs with 10 goals. “Obviously, we would have liked to have gotten a couple of power play goals, but they defended really well. We’ve got to find a way to score - it’s playoff time, there’s no excuses, but they did a really good job.”

And that is the difference in the playoffs. Dallas was able to get past Vegas and Colorado by coming up with the right play at the right time. Edmonton won this series by doing the same thing.

“Hockey is hard,” Seguin said. “You need a lot of things to go right. You need to have that opportunity and we had that opportunity. We went through a gauntlet and beat some really good teams, and we knew we had something special. We lost to a team we thought we could beat and sometimes that’s the playoffs. Sometimes, it’s that one bounce, that one goal, that one save. It’s why we all love it. It’s why this is the hardest damn trophy to win.“

Tyler Seguin speaks to the media after Game 6

The traditional handshake line featured some emotional moments. It could have been the last one for 39-year-old Joe Pavelski, who may retire this summer after 1,533 NHL games (regular and postseason combined). No decision has been announced, but there was a definite understanding Sunday could have been the veteran’s last game.

“Tonight, for me, Joe Pavelski, five shots on net, he turned the clock back,” said DeBoer, who also coached Pavelski in San Jose. “I don’t know if it will be Joe’s last game or not, but it’s been an absolute privilege to coach a guy like that. Our young players are all better for having been around a guy like that.”

Both Benn and Seguin were emotional when asked about Pavelski.

“Can you not ask about Joe . . . ,” Seguin said. “Since day one, he’s meant everything to our group; on the ice, off the ice, he’s just an amazing person to have in here.”

Benn echoed those sentiments.

Asked what Pavelski meant to him, he said, “The world. All-time teammate, person, great leader, good friend.”

And that’s one of the reasons the loss was so frustrating and disappointing. Not only did the Stars feel they should have won the game, they had a belief that they could have won it all this year.

“You never want to go out,” Benn said. “We feel we probably deserved a better result there, but that’s hockey. They got one more goal than we did.”

This story was not subject to the approval of the National Hockey League or Dallas Stars Hockey Club.

Mike Heika is a Senior Staff Writer for DallasStars.com and has covered the Stars since 1994. Follow him on Twitter @MikeHeika.