Ben Bishop made 37 saves in his first start this season for Tampa Bay, then denied each of Chicago's three shooters in the tiebreaker. Filppula beat Corey Crawford into the right side of the net in the first round.
Tampa Bay looked listless before Martin St. Louis and Teddy Purcell scored midway through the third period, tying it at 2. It was the first victory of the season for the Lightning, who opened with a 3-1 loss at Boston.
Patrick Kane and Brandon Saad scored in the second for Chicago, which outshot Tampa Bay 39-16 in the first game between the teams since the Lightning's 5-4 overtime victory on Nov. 4, 2011. Jonathan Toews, Kane and Marian Hossa each were unable to solve Bishop in the shootout.
The Stanley Cup champion Blackhawks were coming off a 6-4 victory over Washington in their season opener Tuesday. They set an NHL record by recording at least one point in each of their first 24 games a year ago, but they wasted an opportunity to earn two points against the Lightning.
Tampa Bay failed to register a shot on goal before Nate Thompson was stopped by Corey Crawford 1:22 into the second. It had only six shots at the end of the period, compared to 25 for the Blackhawks.
The scoreless first was the first time an NHL team had failed to register a shot on goal in a period since Edmonton on March 3 at Minnesota, according to STATS. It was the third such period for Tampa Bay in franchise history, and only the second time Chicago had pulled off the feat, including the first period of a scoreless tie against Detroit on Dec. 4, 1946.
The Blackhawks grabbed the lead when Kane jumped on a loose puck on the right side of the net and punched it in just 59 seconds into the second. Saad then made the most of a big break midway through the period.
With Tampa Bay center Alex Killorn in the penalty box for hooking, Blackhawks defenseman Nick Leddy tried to dump the puck in behind the Lightning net. Bishop went back to get it, but it took a weird bounce off the boards and slid toward the front of the goal.
Saad came in from the left side and knocked it in before Bishop could get into position, making it 2-0 at 9:14.
The two-goal advantage looked as if it would be more than enough before Tampa Bay suddenly woke up midway through the third.
First, St. Louis beat Crawford with a backhander on a rebound with 9:52 remaining. Then, Purcell sent a wrist shot over Crawford's right shoulder for a tying power-play goal just 1:43 after Tampa Bay's first score.
After beginning with the Stanley Cup finalists from a year ago, Tampa Bay is at Buffalo on Tuesday night before beginning its home schedule on Thursday against the Panthers.
NOTES: Tampa Bay also went without a shot on goal in the second period of games against Philadelphia on Nov. 22, 1999, and Phoenix on Dec. 2, 1999. ... D Michael Kostka and F Ben Smith each made their season debut for the Blackhawks. Ds Sheldon Brookbank and Michal Rozsival and F Jimmy Hayes were scratched. ... Ds Keith Aulie and Mark Barberio and C Tom Pyatt were the scratches for Tampa Bay. ... Linesman Mark Shewchyk left the ice midway through the third period due to a leg injury that stemmed from a collision along the boards. The remaining three officials finished the game.