Dvorak, with a stick blade slashing into his back, deflated the Capitals with a breakaway goal at 10:49 of the second period to snap a scoreless tie.
Burke made 26 saves as the Panthers moved within a point of eighth-place Boston in the Eastern Conference.
Florida also moved ahead of the New York Rangers in the five-team race for the Eastern Conference's final playoff berth. The victory was a big turnaround for the Panthers, who were shut out 2-0 at Boston on Tuesday.
Peter Bondra scored with 6:38 to play for the defending conference champion Capitals, who were left in truly desparate straits by the defeat. Washington is 1-3 on its current home stand, which includes losses against fellow contenders New York and Florida.
After half a game of scoreless hockey, Oleg Kvasha won a faceoff just inside Florida's blue line. Dvorak took the puck and sprinted away, outskating defenseman Brendan Witt for a solo run at goaltender Olaf Kolzig.
Witt made a vicious slash at Kvasha just as the forward was scoring his goal, then vented his anger more by taking a two-handed swing into the glass not far from Kvasha's head. Witt received a two-minute penalty for unsportsmanlike conduct and a 10-minute misconduct, further depleting the Capitals' already thin defensive corps.
The Caps killed off the penalty and outshot Florida 27-20 but their few genuine chances the rest of the way were offset by largely uninspired play and good defense by the Panthers, at least until Bondra's late goal. Washington fell to 0-27-1 when trailing after two periods.
Ray Whitney's goal was the clincher, coming 5:17 into the third period after Scott Mellanby picked off Calle Johansson's pass in the corner. Whitney took the feed from Mellanby to score the goal, then had another one disallowed with 10:53 to play because of a player in the crease.
Bondra finally gave the Capitals some life with his 31st goal of the season, a one-timer from just outside the right circle. But the final assualt, with Kolzig pulled in the final minute, failed to produce a clean shot at the goal.
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