Braden Holtby made 34 saves and Ovechkin scored the second of three goals by the Capitals in the first 17 minutes of a 5-2 victory over the Predators on Saturday night, extending Nashville's losing streak to five games.
Troy Brouwer and Eric Fehr also scored for the Capitals. Ovechkin got his 22nd goal this season and ninth on the power play, both NHL highs. Nicklas Backstrom and Mikhail Grabovski each had two assists.
Schmidt, a 22-year-old rookie, got his first NHL goal. Alzner, a six-year veteran, got his first career goal in 148 home games.
"That's what I'm here for. I'm a game-closer. The finisher, they could call me," joked Alzner, who has six career goals and said he had been aware of his failure to score at home since his rookie season.
In the dressing room, Schmidt had a wide smile as he posed for pictures with the puck.
"I've been used to previously having more points," he said. "Being a first-year defenseman, it's going to come."
Nashville defenseman Shea Weber, playing for the first time since getting hit in the face by a puck on Nov. 28, also had an assist.
"We couldn't find a way to score. Obviously, we didn't have many power plays tonight [one] to try and build momentum," Weber said. "Same story, we've just got to find a way to put it in."
Washington took the lead after Nashville goalie Marek Mazanec (28 saves) failed to control a slap shot from the wing by Fehr. Battling a Predators defender, Brouwer got enough of his stick on the puck to trickle a backhand past Mazanec for his sixth goal.
Less than 7 minutes later, Ovechkin scored 3 seconds into a power play, firing a slap shot off a faceoff feed from Backstrom.
Alzner made it 3-0 with his first regular-season goal since Feb. 12. In addition to winning a draw that led to the goal, Backstrom played a role on the shot. He obstructed the view of Mazanec, slow to react on a drive from the defenseman that came from just inside the blue line.
Alzner has often been criticized for a lack of aggressiveness with the puck.
"When it's his turn, we need a goal from him, or a good play, and we got one tonight," Washington coach Adam Oates said. "Every little touch he makes benefits us big picture."
The teams traded goals early in the third period. After Schmidt scored the first of his career, Nystrom retaliated 23 seconds later in transition.
With 5:10 left, Fehr got the clincher, scoring for the fourth time this season. He has six points in six games.
"I think we were getting a little bit too fancy, looking for that perfect tap-in, backdoor play," Fehr said. "We talked before the game [and said], just get pucks to the net and crash and get some rebounds and try to get a greasy one. It worked out."
Ex-Capital Matt Hendricks faced Washington for the first time since he was signed by Nashville as a free agent. ... The losing streak is the longest for Nashville since it dropped eight straight in April. ... Nashville's frustration showed as Rich Clune picked a fight with Washington rookie Tom Wilson, but got the worst of it after his helmet came off in the tussle. ... With the 136th power-play goal of his career, Ovechkin moved within one of Peter Bondra's franchise record. ... The goal for Alzner was his first in 148 regular-season games at home.