Purcell got his fifth goal, from the slot, after a turnover by Kyle Quincey behind the Detroit net.
Zetterberg tied the game with 4:12 left in the third on a slap shot from the right circle for his second of the game and 10th goal of the season.
Stamkos gave the Lightning a 2-1 lead with 7:10 left in the second period. He beat Howard with a wrist shot from the right circle.
Stamkos' 14th goal came four seconds after Tampa Bay had killed off 5:24 of consecutive Detroit power-play time with 36 seconds of it a two-man advantage.
In all, the Lightning had only been at even strength for 13 seconds of the previous 7:37 when the final penalty ended.
The goal extended Stamkos' goal streak to six games and his points streak to eight.
Zetterberg opened the scoring 3:51 into the game when he intercepted Stamkos' pass in the slot and put a wrist shot past Bishop.
Malone tied it at 1 with 9:49 remaining in the opening period when he put in the rebound of his own wrap-around attempt. It was Malone's second goal.
NOTES: It was Tampa Bay C Valtteri Filppula's first game against his former team after signing a five-year, $25 million free agent contract with Tampa Bay in July. Filppula spent his first eight NHL seasons with the Red Wings. ... Detroit C Stephen Weiss didn't play because of a sore groin and RW Johan Franzen missed his fourth game with an upper body injury/illness. ... Lightning GM Steve Yzerman played his entire 22-season Hall of Fame career with the Red Wings and was the Detroit captain for the last 19 seasons. . The Red Wings recalled C Luke Glendening and D Xavier Ouellet from their AHL affiliate in Grand Rapids earlier on Saturday.
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