The Capitals ended a three-game skid and wound up with a 2-2-1 record on their five-game homestand.
Washington began the night two points behind Winnipeg for the eighth and final post-season spot in the Eastern Conference.
Tampa Bay held a 2-1 lead heading into the third period but Marcus Johansson tied it with 3:58 to go in regulation.
Ovechkin then beat Dustin Tokarski with the game-winning shot in the first career start for the Tampa Bay goaltender. Tokarski gave up three goals on 32 shots.
The win went to Tomas Vokoun, who made 24 saves.