Ovechkin's four goals gave him an NHL-best 26 for the season and was the third such performance of his stellar career.
His fourth of night came with 32.4 seconds left in the third period, completing a comeback in which Washington rallied from a 3-0 deficit.
Troy Brouwer scored in the fifth round of the shootout and Phillipp Grubauer stopped Ondrej Palat on the Lightning's final try to seal the Capitals' third consecutive win and fifth in six games.
Nicklas Backstrom supplied a goal and four assists.
Grubauer turned aside 32-of-34 shots after talking over in net from Braden Holtby, who was pulled in the first period after coughing up three goals on just eight shots.
Tyler Johnson and Martin St. Louis each notched a goal and two assists for the Bolts, who dropped a sixth straight on the road.
Palat provided a goal and an assist and Ben Bishop stopped 26 shots.