Ryan Malone scored the deciding goal early in the third period for the Lightning, who won for the third time in their last four games.
Malone scored at 6:45 of the third period to put Tampa Bay ahead 4-2 and give Smith enough of a cushion to withstand Kovalchuk's goal at 16:50.
Kovalchuk assisted Antropov on the game's first goal at 11:48 of the first period.
The Lightning tied the game less than 2 minutes later on Downie's goal and went ahead 2-1 at 4:44 of the second on Mike Lindin's first career goal.
After Atlanta tied the game 2-2 on Reasoner's goal, Alex Tanguay chipped in a goal that made it 3-2 at 5:04 of the third.