Tarnasky sent a low shot between Brent Johnson's pads to help the Lightning improve to 10-1 in shootouts. It was Tarnasky's first career shootout attempt.
Washington is 1-10 in shootouts.
With 3:18 left in regulation, Martin St. Louis scored to give the Lightning a 4-3 lead, taking a pass from Vincent Lecavalier, getting a step on Milan Jurcina in the slot and poking in a falling-down shot past Johnson.
But Alexander Semin tied it with 10.9 seconds left, on a left-post attempt that went in off defenceman Nolan Pratt's stick. The Capitals had pulled Johnson and were on a two-man power-play for a 6-on-3 advantage.
Vaclav Prospal, Ruslan Fedetenko and Lecavalier also scored in regulation for the Lightning in the first game of a six-game trip. Tampa Bay has points in 10 of 11 road games, going 9-1-1 in that span.
Shaone Morrisonn, Alex Ovechkin and Matt Pettinger also had goals in regulation for the Capitals, who have lost three straight and eight of nine.