Washington took a 2-1 lead 1:51 into the middle period on a goal from Eric Fehr, but Tampa Bay dominated the rest of the way to wrest home-ice advantage away from the Capitals.
The second game of the Eastern Conference semifinals will be played Sunday in Washington.
Steve Downie tied it with 3:43 to go in the second and Stamkos then produced a power-play goal with 32 seconds left in the period. He scored on a backhander that beat Michael Neuvirth.
Stamkos produced 45 goals during the regular season, second in the NHL behind the 50 Corey Perry tallied for Anaheim. This was the third tally of the playoffs for Stamkos.
It remained a 3-2 contest until Adam Hall scored into an empty net with 42 seconds left in the third.
Dwayne Roloson made 26 saves to get the victory. The 41-year-old Roloson, who is playing for his sixth NHL team, was a key factor in Tampa Bay's comeback from a 3-1 deficit in the conference quarterfinals against Pittsburgh, a series the Lightning captured with a 1-0 Game 7 victory.
Neuvirth stopped 20-of-23 shots.