New Jersey moved to the brink of clinching a playoff position, needing only one point from its final four games or a loss by Washington to nail down a post-season spot.
Tampa Bay took a 3-2 lead when Ryan Malone scored midway through the second period to complete a hat trick, but Kovalchuk tied it 3 minutes later and Marek Zidlicky put the Devils in front on the power play with 5:23 to go in the middle stanza.
Martin Brodeur gave up four goals on just 20 shots, but still picked up the 653rd win of his career. He needs one post-season triumph to reach 100 in his 19 NHL seasons.
Dwayne Roloson was removed from the game after giving up six goals on 34 shots. Sebastien Caron cleaned up and stopped all four shots he faced.