George McPhee — the general manager who selected Ovechkin and constructed Washington’s spine, from Backstrom to John Carlson to Kuznetsov to Holtby — moved on to assemble in Vegas the most successful first-year franchise in major North American sports history. Vegas romped to a Pacific Division title and burned through the Western Conference bracket, losing only three times in its first 16 playoff games.

Facing elimination on Thursday, the Golden Knights confronted their predicament with defiance. Opening their pregame festivities, a video implored fans not to give up — if the Boston Red Sox in 2004 and the Cleveland Cavaliers in 2016 and the New England Patriots in 2017 could overcome imposing deficits, then so, they hoped, could Vegas.

“After you get past the losing here, you can look back and be pretty proud of the group in here and what they’ve done getting into this community and the city after what happened, and the run we went on,” said Vegas defenseman Deryk Engelland, referencing the mass shooting here on Oct. 1 that strengthened the Golden Knights’ bond with the city. “Everyone had us pegged to not make the playoffs. To be standing here today, as bad as it feels, you’ve got to be proud of the group in here.”

In contrast to these Golden Knights, the Capitals compiled in their expansion season of 1974-75 what is still regarded as the worst season in league history: an 8-67-5 record worth 21 points. The franchise matured into a perennial contender, and for more than a decade Washington has been one of the N.H.L.’s top teams, winning its division eight times in 11 seasons and making 10 postseason appearances over all in that span.

Each of those playoff forays had been defined, in one way or another, by calamitous defeat: to eighth-seeded Montreal in 2010; in seven games to the Rangers in 2012, 2013 and 2015; in consecutive series to Pittsburgh in 2016 and 2017, despite finishing with the most points in the league both seasons. In the Ovechkin era, the Capitals have twice bungled three-games-to-one leads, and before vanquishing Tampa Bay last month, they had lost seven of 10 Game 7s.

“Years past, it felt like we were like, ‘We’re going to win this year,’” forward T.J. Oshie said. “This year, it was more so like, ‘We’re going to work this year, and we’re going to outwork teams. We’re going to play longer than they are.’ And we did that. We did that throughout the whole playoffs.”

Oshie spoke in a stream of consciousness, at one point apologizing for rambling. When he finally found his family on the ice, he cradled one of his daughters and told her, “We won the trophy!”