LAS VEGAS — The Caps’ cup, their Stanley Cup, runneth over, at last.

For decades, the Washington Capitals and their fans could have filled the huge Stanley Cup with their tears of frustration. Now they can fill it with tears of joy.

Captain Alex Ovechkin, the MVP of the playoffs, was the first to skate around the rink, carrying the Cup over his head after a 4-3 victory in Game 5 of the finals over the Vegas Golden Knights. Then the Great 8 handed the Cup — or in the Caps’ case it should be grail — to Nicklas Backstom, his buddy and running mate for the past 11 years.

Skating side-by-side with Backstrom, Ovechkin bellowed, “Yeah, baby!” And those words, yelled through a gaptoothed grin, could stand for the emotions of countless delirious, delighted and oh, so relieved Caps fans.

The concept that “justice too long delayed is justice denied” can be traced back centuries. But among its many applications, the Capitals, and especially the players of the Ovi era, deserve to have it applied to them. This Cup is justice delivered — just in time, when many in the NHL thought they had squandered their last chances — by the Caps’ own determined hands.

A diving, fully extended, flick-of-the-wrists goal by Devante Smith-Pelly, flying in front of the crease parallel to the ice, tied the score at 3 with 10:08 left to play. That brilliant play, set up by a shot by Brooks Orpik, stunned the Golden Knights and their raucous crowd.

Just 2:31 later, the Caps got their winning goal from Lars Eller. Brett Connolly ripped a slap shot at goalie Marc-Andre Fleury, a Capitals nemesis in his Pittsburgh days. For an instant, no one knew where the puck was — except Eller. The forward was standing entirely behind Fleury, gazing down at an almost motionless puck sitting just inches behind the goalie’s rear end and also inches from the goal.

Mostly, it was a result of Caps skill, a ton of Caps pressure throughout this game and this series. But, for once, perhaps just an ounce of good fortune, decades past due, was in play, too. Eller tapped it in, shorter than a gimme putt.

Delirious Caps, including 22-year-old rookie Jakub Vrana, who scored the first goal of the game on a deft breakaway, took their turns as Stanley Cup hoisters. Back in D.C., delighted yet almost disbelieving Caps fans filled streets all over town with their delight.

Yet there was something else besides Washington’s first major pro sports championship in 26 years and the Caps’ first Cup in their 44 years of existence that was being celebrated. Everyone who cared for the Caps or for their fans understood that there was something else that “we” wanted.

That desire, marinated in a third of a century of sports misery, was for a final score, a season-ending outcome that reflected simple sports fairness.

Please, can we have a deserved resolution to an old ache? Can we, at last, expunge the Caps’ ugly, only partially deserved reputation as gifted, inexplicable chokers? And can we replace it with an image of undiluted joy?

Or, if you prefer, a memory of a gray-haired, gloriously relieved Ovechkin, beaming at thousands of Caps fans who roared throughout this game, who will now always be remembered as a champion.

In the end, the Caps not only had their long-deferred Cup but an entire first-class organization. A consistent winner since the 1982-83 season and the best regular season team in the NHL over the past 10 years, Washington would ultimately be remembered for its gifts, for its ability to keep hoping and battling. And not solely for 35 years of incessant, almost sadistic postseason disappointment and ill luck.

In the past decade, the Caps lead the NHL in regular season points with 1,019, ahead of the Penguins (1,008) and the Chicago Blackhawks (988). Nobody else is close. In those 10 years, the Penguins and Blackhawks have won six Cups — three each. In the NHL’s luck-laced playoffs, no one team, no matter how good, is assured of winning a title in any particular year. But sustained excellence for so long, as Pittsburgh and Chicago illustrate, is usually rewarded.

And should be because it is deserved. Now, add the Capitals.

In a two-month span, through four playoff series, the Caps relived every sad, unlucky or mortifying event that had dogged their franchise since the day it was born. Name a hockey sin of which the Caps have been accused, and as they defeated the Columbus Blue Jackets, Penguins, Tampa Bay Lightning and, finally, the Golden Knights to win the Cup, they have atoned for it.

Against Columbus, the Caps lost the first two games at home — instant crisis — and, in desperation switched back to Braden Holtby, their recently benched goalie, for stability. They won four straight. Next, they beat their nemesis — the Penguins with Sidney Crosby. And the Caps did it despite a Game 1 loss at home, an injury to Backstrom and a three-game suspension of valuable Tom Wilson. They even closed out the series with an overtime goal in Game 6 in Pittsburgh by Evgeny Kuznetsov, the team’s breakout superstar of the postseason, who finished the playoffs with 32 points.

Their toughest test was Tampa Bay. By winning the first two games on the road, the Caps set an insidious trap — for themselves. Suddenly, they were favorites, and when they lost the next three games, they were on the brink of recapitulating the team’s most hideous narrative. Could the Caps really blow a two-game lead, either 2-0 or 3-1, for the 11th time in 34 years?

The Caps, who supposedly never rise to the occasion against tough foes for high stakes, dominated the Lightning, outscoring Tampa Bay 7-0 in the last two games.

Even in this final game, the Caps had demons to dispel. As Las Vegas overcame Capitals leads of 1-0 and 2-1 to take leads of 2-1 and 3-2, those dismal thoughts may have haunted some.

But now we know. History can never be erased, but it can be redrafted by subsequent events — even gloriously so.

Perhaps this night in Sin City will be the one when Caps faults are forgiven. Maybe this is the fulcrum event when Washington finally allowed itself to believe that optimism in sports is not a disease, that rooting for the Capitals can be as great a delight as it has been a burden and that this idea of investing many years in chasing and capturing a championship is an experience that a team and an entire city can share and remember. For as long as they want.

Shed a tear. But this time of joy. Drop it, if only in your imagination or in your heart, into the Stanley Cup, at last.