Tom Wilson had a goal and an assist, and Dylan Strome scored for the Capitals (34-36-9), who have lost six in a row (0-5-1). Lindgren made 33 saves.
"It's unfortunate the way it ended," Washington coach Peter Laviolette said. "I thought the guys played hard tonight. They competed hard, they were invested, they were physical, they were skating. They had to overcome a lot of things on the ice."
Tkachuk (16:28 of the first period) and Reinhart (2:28 into the second period) each had deflection goals disallowed because of high-sticks, keeping the score 0-0.
Strome gave the Capitals a 1-0 lead at 8:51 of the second when he lifted the rebound of Wilson's shot from the slot over Lyon's left pad.
Carter Verhaeghe tied it at 1-1 at 10:56 with a slap shot from above the right circle for his team-leading 41st goal of the season, and Aaron Ekblad gave Florida a 2-1 lead with a power-play goal, scoring on a rebound at 13:47.
Wilson tied it 2-2 at 16:05 when he drove to the net on the rush to redirect Conor Sheary's pass from the left wing over Lyon's left shoulder for his 10th goal of the season.
"I think it was a better game," Wilson said. "We probably deserved kind of a better fate, but just didn't get it done."
Tied after two periods and knowing the Penguins won earlier in the day, the Panthers stepped up the pressure in the third and outshot the Capitals 18-3. But it took until there was a minute remaining for Tkachuk to score go-ahead goal.
Reinhart scored into an empty net with 40 seconds left for the 4-2 final.
"I love that third. I love the way we came out," Reinhart said. "You know sometimes when you're playing well, it takes a while to finally break through. I think just some relief there that there's not a bounce that goes the other way so. Obviously, a big goal by [Tkachuk] to put us ahead."