Hedman fired a wrist shot that just made it inside the left post past Ondrej Pavelec. Mike Smith then stopped Evander Kane to remain perfect in the shootout and give the Lightning their second straight win.
The only goals in regulation came from Atlanta's Rich Peverley, shorthanded in the first period, and Tampa Bay's Sean Bergenheim in the second.
Smith made 27 saves in the victory and helped the Lightning move into a second-place tie with Atlanta in the Southeast Division, two points behind Washington.
Pavelec stopped 30 shots in what was Atlanta's second loss in six games.