Sitting in the eighth and final post-season spot in the Eastern Conference, the Capitals had a chance to gain some breathing space but remain in a vulnerable position, just two points ahead of ninth-placed Buffalo with two games remaining.
"We're fighting for our play-off lives," Washington wing Jason Chimera said. "It means a lot playing for something this time of year. You want to be in that hot seat."
In the late game, Western Conference contenders Los Angeles earned a 2-0 shutout over Edmonton to take a two-point edge over Phoenix for (39-27-13) top spot in the Pacific Division and the third seed.
The Kings (40-27-13) end their regular season with a home-and-home series against San Jose beginning on Thursday.
In Tampa Bay, Washington was on the verge of at least earning a point as the game reached its closing stages.
The visitors were tied 2-2 late in the third, but Stamkos buried the game-winner from in front of the goal with 1:03 left and added an empty-net score in the last seconds to run his NHL-leading goal total to 58 on the season.
It was a brutal loss for the Capitals (40-32-8), who also could have moved within a point of Southeast division-leader Florida but are now in danger of missing the playoffs completely.
Alexander Semin put Washington ahead 1-0 in the second period before Teddy Purcell and Victor Hedman responded about five minutes later to put the Lightning (37-35-7) in command.
Chimera came up with the game-tying score at 16:43 in the third but it was quickly erased as Stamkos seized the spotlight and sent the Capitals home sulking.