Ryan Nugent-Hopkins' wrist shot with less than four minutes remaining saw the Oilers to their first win in six.
Despite finally clinching their first win, the Oilers stay bottom of the Pacific division, while the Lightning sit fourth in the Atlantic standings.
Ben Bishop stopped 20 shots for Tampa Bay and Ben Scrivens made 22 saves for the victors.
It took until the second period for the first goal to come at Rexall Place and the Lightning took the lead through Brian Boyle.
Taylor Hall would help turn the game for the Oilers, assisting Justin Schultz's wrist shot before scoring to put his team ahead.
Brett Connolly tied the game with just over six minutes remaining before Nugent-Hopkins' goal saw the Oilers to victory.