New Jersey won its third in a row with Martin Brodeur making 17 saves to collect the 661st regular-season victory of his career.
Kovalchuk snapped a 1-1 tie with 2:26 to play in the second period and the Devils took charge in the third when Andy Greene and Patrik Elias scored 14 seconds apart as the game was entering its final 5 minutes.
Both third-period goals came on the power-play after Marc-Andre Bergeron was penalized for verbal abuse of an official.
Adam Enrique scored New Jersey's first goal while Nate Thompson and Matt Carle tallied for Tampa Bay. It was the Lightning's third straight loss after they had won five in a row.
The setback went to Anders Lindback, who gave up all four goals on 27 shots.