Richard Panik had two goals and Tom Wilson also scored for Washington (34-11-5), which had won four in a row. Braden Holtby made 19 saves.
"However it happens, to go into the third period with a lead and lose it, and especially how we lost, it was disappointing for sure," Capitals coach Todd Reirden said.
Panik gave Washington a 1-0 with a wrist shot from the top of the right circle at 9:28 of the first period.
Jarred Tinordi's first NHL goal tied it 1-1 with a one-timer from beyond the left circle at 10:15 off a pass from Filip Forsberg.
"Obviously, it was a good pass by [Forsberg]; he kind of left it on a platter for me and I shot it," Tinordi said. "I didn't quite see it go in. I just saw him coming in for the big hug, so that was nice."
Mikael Granlund gave the Predators a 2-1 lead with a power-play goal at 12:52, and Rocco Grimaldi made it 3-1 when he slid the puck between Holtby's pads at 16:02.
Ovechkin scored No. 693 at 16:26 of the first period after Saros was unable to handle Nick Jensen's dump-in from center ice to make it 3-2.
The Capitals tied it 3-3 when Nick Bonino accidentally put the puck in his own net at 6:01 of the second. Panik was credited with the goal.
Wilson gave Washington a 4-3 lead when he knocked in a rebound at 11:06 of the second period during a power play.
Johansen tied it shorthanded when he intercepted Holtby's pass at center ice, skated into the Capitals zone and scored with a wrist shot from the right circle.
"Obviously you want to win it, but the second [goal] was, can we come out of this game win or lose with some identity of pushback and getting through those situations?" Predators coach John Hynes said. "I really like the fact that we were able to stay with it and get rewarded for it."
It was the fourth shorthanded goal the Capitals have allowed in their past seven games.
"I just didn't see him there," Holtby said. "Just bad awareness there. Trying to do too much, I guess. … Just a play that can't happen. That's what killed us."
