Patric Hornqvist scored two goals and Shea Weber added another for Nashville, which has won two of three. It was the Predators' first shootout win since Jan. 16 at Philadelphia and their second of the season against eight losses.
"I thought we played better today than yesterday," Washington coach Adam Oates said. "There were moments where we didn't execute, but I thought that as we got into the game, we played better."
Nashville goaltender Carter Hutton had 23 saves, and then denied all three Washington shooters in the tiebreaker.
Weber gave Nashville a 3-2 lead at 8:23 of the third.
Matt Cullen carried the puck into the Washington zone on the right side. Just across the blue line, he left a drop pass for Mike Fisher, who found Weber at the top of the right circle. The captain then ripped a slap shot high to the stick side of Capitals goaltender Jaroslav Halak for his team-leading 21st goal of the season.
"The way (Weber) shoots the puck, it is pretty dangerous every time," Cullen said.
Washington then used a power play to tie the game again with 7:40 to go. Backstrom's snap shot from above the right circle beat Hutton for his 14th of the season.
The Capitals went 2 for 4 with the man advantage. They began the day with the league's best power play, clicking at a success rate of 23.8 percent.
"I thought we deserved the game," Nashville coach Barry Trotz said. "I thought 5-on-5, we were much better than they were tonight. We managed the pucks well."
Nashville did not get a power play in the game.
Brouwer scored the game's first goal at 4:17 of the opening period.
From the left corner, Backstrom found Mike Green in the right circle. Green made a quick backdoor pass to Brouwer, who beat Hutton with a one-timer.
Hornqvist responded at 5:55 of the first.
Fisher won a faceoff at the right dot inside the Washington zone back to Weber, where his slap shot was stopped by Halak. On the left side, Hornqvist grabbed the rebound and put it past a fallen Halak for his 100th career NHL goal.
"He's been great for us all year," Hornqvist said of Weber. "He's our leader. He's our best defenseman on both sides of the puck and obviously he has a big shot. He showed it tonight."
Hornqvist struck again at 18:30 of the first, when a Backstrom turnover at the Washington blue line sent him in on a breakaway. From in close, Hornqvist beat Halak just under the crossbar for his 17th of the season.
Just seconds before Hornqvist's goal, a sliding Weber knocked the puck away from Alex Ovechkin, who was skating down the right side of the Nashville zone, denying the league's leading scorer of a solid chance.
"There aren't enough games left to be too positive about only getting one point," Washington defenseman John Carlson said. "Yeah, it is a point, but it's only going to make our road harder."
Green played in his 500th career NHL game. He is the 10th defenseman in Capitals history to reach that mark with the team. ... Brouwer has points in his last three games. ... Fisher has six points in his last six games.
Copyright by STATS LLC and The Associated Press
NASHVILLE, TN - MARCH 30: Carter Hutton #30 of the Nashville Predators hands a puck to 4-year-old Carli Camelio...