Parise also scored a power-play goal 1:13 into the third period that brought about a 2-2 tie and contributed to the Devils' fifth win in six games. Kevin Weekes made 39 saves to run his winning streak to five games.
Both Tampa Bay goals came from Ryan Malone in the first period, but the Lightning let the lead get away and lost for the eighth time in 10 outings. Karri Ramo made 38 saves.
Vincent Lecavalier scored for Tampa Bay to open the shootout, but his goal was matched by Parise.
After Weekes turned aside a shot by Vaclav Prospal, Elias produced the deciding goal. Ryan Malone had a chance to keep the Lightning alive, but he could not get a shot past Weekes.
Weekes also stopped a breakaway bid by Evgeny Artyukhin with 7 minutes to play in regulation that kept the game tied.
Notable deaths of 2014 [PHOTOS]