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The Washington Capitals didn't have to go far to get the party started Thursday night ... 'cause the Stanley Cup champs turned all the way up in a Vegas nightclub ... and TMZ Sports has the video.

Alex Ovechkin and company hit up Hakkasan at the MGM Grand -- where the entire team, coaching staff, management, owners and fans went berserk to celebrate the victory over the Vegas Knights.

DJ Tiesto was in charge of the music and invited the team over to the VIP area behind the booth to party with the cup.

We're told team management paid the entire bill for all Caps players -- which ended up around $100,000 ... and that covered all of the champagne, beer and liquor the guys poured into the cup.

Early Friday morning, President Trump tweeted at the team -- "Congratulations to the Washington Capitals on their GREAT play and winning the Stanley Cup Championship."

"Alex Ovechkin, the team captain, was spectacular - a true Superstar! D.C. is popping, in many ways. What a time!"