Both sides scored three times during the first four rounds of the shootout, but that rash of gold scoring was followed by 11 consecutive failures before Kopecky fired a wrist shot past Grubauer.
All the scoring during regulation came in the second period with Florida twice taking the lead. Dylan Olsen and Aleksander Barkov scored for the Panthers and Nicklas Backstrom and Joel Ward had goals for the Capitals.
Florida won for the fourth time in five games and ended an eight-game losing streak against Washington.
Scott Clemmensen earned the win with 23 saves. Grubauer stopped 39 shots.