Four players from each team had scored in the first 12 rounds, and Salo gave Tampa the advantage when he lifted a backhander past Evgeni Nabokov.
Ben Bishop then preserved the win as he stopped Johan Sundstrom's offering.
Ryan Callahan scored both goals while Bishop turned away 25 shots for the Lightning, who had dropped their previous two.
With the win, Tampa now has 89 points and sits in the third spot in the Atlantic Division, two back of Montreal.
Matt Donovan posted a goal and an assist while Ryan Strome gave a goal for the Islanders, who had won four of six coming into the game. Nabokov stopped 29 shots.
After a scoreless first period, the Lightning grabbed a 1-0 lead just 29 seconds into the second as a big blast from the right point by Radko Gudas was tipped in by Callahan, who was parked in front of the net.
The Islanders, though, tied the game just 1:04 later as Strome's chip shot from the left side fooled Bishop.
New York then took a 2-1 lead near the midway mark of the second as Frans Nielsen's backhand pass from the left of the net went out to the slot and Donovan one-timed it home for his second of the season.
Tampa tied the game at 5:50 of the third as Callahan just kept the puck in near the left point and whipped it over to Eric Brewer, who fired a shot on net that saw Callahan clean up the rebound for his 15th of the season.