St. Louis scored twice in the middle frame, erasing 1-0 Devils lead, and added another tally midway through the final period on a tip-in of a Teddy Purcell shot, put the Lightning up 4-2.
They needed that tally when Zach Parise drew the Devils to within a goal on a power-play marker with 2:35 remaining in regulation.
St. Louis also assisted on Matt Gilroy's goal and Mathieu Garon turned aside 19-of-22 shots for the Bolts, rebounded from an 8-1 drubbing by Pittsburgh Saturday and pulled to within six points of the eighth and final Eastern Conference playoff slot.
Martin Brodeur was touched for four goals on 16 shots in the loss.
Swim Week Miami Beach 2014 [PHOTOS]