Richards scored in each of the first two periods and then avoided a poke check from New Jersey goaltender Martin Brodeur, and lifted the puck over the goalie's shoulder for the game-winner in the shootout.
The victory was Tampa Bay's sixth in its last seven games while the Devils saw a three-game winning streak end.
Jay Pandolfo scored a goal 1:58 into the contest for New Jersey but Richards tied the score about two minutes later by out-skating two defenders and then beating Brodeur.
He made it 2-1 in the second period when he knocked in a rebound with four seconds left on a power-play situation.
New Jersey forced overtime with a power-play goal from Paul Martin with 7:10 left in regulation. It was Martin's first goal of the season.