Craig Smith scored, and Darcy Kuemper made 29 saves for the Capitals (34-33-9), who have lost six of seven (1-4-2), including a 2-1 shootout defeat against the New York Islanders on Wednesday.
Washington is seven points behind the Pittsburgh Penguins, who have one game in hand, for the second wild card into the Stanley Cup Playoffs from the Eastern Conference. The Penguins defeated the Nashville Predators 2-0 on Thursday.
"Our first period was tough, that wasn't the way we wanted to start," Capitals coach Peter Laviolette said. "We had a much better second period and gave ourselves a chance going into the third, but the start was no good. We weren't quick enough. I liked the way we played in the second period, but at the end of the night we lost the game."
Killorn gave Tampa Bay a 1-0 lead with a power-play goal at 9:20 of the first period, a redirection of a shot from Mikhail Sergachev.
Stamkos extended the lead to 2-0 at 19:57 with a shot from a high slot off a face-off win by Pierre-Edouard Bellemare.
"We want to be a hard team to play against," Tampa Bay forward Anthony Cirelli said. "We want to limit chances and opportunities, especially 5-on-5 chances. We've been great over the past few games and we have to keep building on that. We have to bring it up to another level."