Game 7 is at Tampa Bay on Wednesday.
[WATCH: All Lightning vs. Capitals highlights | Complete Lightning vs. Capitals series coverage]
The winner will face the Vegas Golden Knights in the Stanley Cup Final, which begins Monday. It will start in Tampa Bay if the Lightning win and in Las Vegas if the Capitals win.
"We can build some confidence after this win and go down there to win Game 7," said center Nicklas Backstrom, who had two assists.
Video: TBL@WSH, Gm6: Holtby posts 24-save shutout in Game 6
T.J. Oshie scored two goals and Devante Smith-Pelly had one, and Braden Holtby made 24 saves for the Capitals, who won at home for the first time in the series. They are 4-5 at home in the Stanley Cup Playoffs and 7-2 on the road.
It was Holtby's first shutout of the season, regular season included, and the fifth of his NHL playoff career.
"It's a perfect time," Smith-Pelly said of Holtby's first shutout since April 5, 2017. "He's been great all year. Obviously an up-and-down year for him personally, but the way he's bounced back, he's been amazing all throughout the playoffs, and really, really happy for him."
Andrei Vasilevskiy made 31 saves for the Lightning, who won the three previous games, including Games 3 and 4 here. They are 5-2 on the road in the playoffs and 6-3 at home.
Tampa Bay was shut out for the first time this postseason after scoring at least two goals in its first 15 games, including three or more in 11.
The Lightning haven't scored since Ryan Callahan's goal 33 seconds into the second period of Game 5, a drought of 99:27.
The Capitals outshot the Lightning 34-24 and outhit them 39-19. It was Washington's most physical game of the series.
"We were no good," Tampa Bay coach Jon Cooper said. "We didn't play with near the desperation they did. Was it a fairly even game? There's no question. But what were the hits, 39-19? Somebody was engaged and somebody wasn't. That's a choice."
Oshie gave the Capitals a 1-0 lead with a power-play goal off a one-timer from above the right hash marks at 15:12 of the second period. Backstrom threaded the pass to Oshie from the right face-off circle.
It was Washington's first power play since 7:08 of the second period in Game 4.
"I think the biggest thing there is No. 8 [Alex Ovechkin] in his office over there," said Oshie, who ended a six-game goal drought with his sixth of the playoffs. "Everything, how teams play us, all depends on where he is and how they want to play him. For me, it's just a matter of a couple feet here and there to try and find a soft area. Whether it's Nick or [Evgeny Kuznetsov], they're able to get the puck into the wheelhouse, and it's up to me to find a hole."
Smith-Pelly made it 2-0 at 10:02 of the third period, scoring on a one-timer from the right hash marks off a pass from behind the net from Chandler Stephenson. It was Smith-Pelly's fourth goal of the playoffs and second of the series. He also scored in Game 2.
Oshie scored an empty-net goal at 19:10 of the third period to make it 3-0.
"They played like it was the last game," Lightning captain Steven Stamkos said. "We played like it might be our last game. The desperation needs to be higher, and I expect it to be."
Video: TBL@WSH, Gm6: Ovechkin does it all for the Caps
Goal of the game
Oshie's goal at 15:12 of the second period.
Video: TBL@WSH, Gm6: Oshie opens the scoring with PPG
Save of the game
Holtby stopping Ondrej Palat at 6:59 of the third period.
Video: TBL@WSH, Gm6: Holtby flashes the leather on Palat
Highlight of the game
Smith-Pelly's goal at 10:02 of the third period.
Video: TBL@WSH, Gm6: Smith-Pelly buries no-look pass
They said it
"You can sit here and say, 'Look at Tampa, they were down 0-2 and they pushed it to a Game 7.' Or you can say, 'They blew a great opportunity to close this series out.' In the end, there is a Game 7. And it's at home. And if you said to me you only have to go 2-2 at home in this series and you're going to win it, we'll take that. We took two on the road, and now we have to make sure we go home and take care of business. Shame on us if we don't." -- Lightning coach Jon Cooper
Video: TBL@WSH, Gm6: Cooper on struggles, looking ahead
"I don't think there's [another] team I've ever had that I'd want to go into a Game 7 with. This team has done a lot of special things this year. It's grown. It continues to do that. What an opportunity going into Tampa. We've won on the road. We've won at home here. We'll see if we can earn the right to keep playing." -- Capitals coach Barry Trotz
Video: TBL@WSH, Gm6: Trotz reflects on Capitals effort
Need to know
The Capitals are 3-7 in Game 7 since 2008 and 4-11 all time, including 1-3 on the road. ... The Lightning are 5-2 all time in Game 7, including 2-2 in the conference final, the wins coming in 2004 and 2015. ... Holtby is the second goalie in Capitals history to have a shutout in a potential elimination game. Semyon Varlamov did it in Game 5 of the 2009 conference quarterfinals against the New York Rangers (4-0). ... Kuznetsov had an assist on Oshie's first goal to extend his point streak to nine games (six goals, eight assists), tying Backstrom's Washington playoff record (2009).
Game 7 of Eastern Conference Final at Amalie Arena on Wednesday (8 p.m. ET; NBCSN, CBC, SN1, TVAS)
Video: Caps stave off elimination, force Game 7 with 3-0 win
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