"That’s another new one for me," Cooper said of the pair of penalty shots.
Tampa Bay outshot the Flyers 5-3 during overtime, and 32-25 for the game.
"There’s definitely a lot of chances out there, and that’s what the league was trying to get out of it," Flyers goalie Steve Mason said. "I think we’re going to see a lot of games ended in overtime just because of how wide open it is."
It was the first NHL game for Philadelphia coach Dave Hakstol, who spent the past 11 seasons coaching at the University of North Dakota. Seven of his teams at North Dakota reached the NCAA Frozen Four.
"I guess I haven’t really thought a whole lot about it," Hakstol said. "I like the way our team played. That’s, I guess, what my focus is. I’m excited to see everybody go out and work hard to play their role and do their job and do it with confidence."
Matt Read and Brayden Schenn scored for the Flyers. R.J. Umberger had two assists.
The Lightning also got a goal from Ryan Callahan. Tampa Bay unveiled its 2015 Eastern Conference championship banner before the game.
Jonathan Drouin picked up two assists as the Lightning improved to 9-0-1 in their last 10 home games against Philadelphia.
After Read beat Bishop from the low slot at 10:14 of the second, Schenn put the Flyers up 2-1 with a rebound power-play goal 1:49 later.
Callahan got Tampa Bay even at 2 on the rebound during a power play with 1:38 left in the second.
Garrison opened the scoring from the blue line when his shot through traffic eluded Mason 1:25 into the second.
Mason kept it a one-goal game less than a minute later by stopping a breakaway shot by Ondrej Palat.
Flyers captain Claude Giroux had a penalty shot with 6:13 left in the first, but sent a soft shot wide of the net after several stick-handling moves.
Notes:Lightning captain Steven Stamkos, in the final season of a five-year, $37 million contract, had three shots but was held without a point. Without a new deal, he could become a free agent after the season. … Former Tampa Bay captain Vincent Lecavalier and C Sam Gagner were healthy scratches for the Flyers. … Tampa Bay C Cedric Paquette (foot) is practicing but didn’t play.