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Niklas Kronwall, Justin Abdelkader, and Darren Helm also scored for the Red Wings, who have won three in a row and seven of 11.
Tampa Bay got goals from Victor Hedman and Steve Downie. The Lightning are 5-3-1 during a 12-game homestand.
After wasting a two-goal lead, the Red Wings rebounded on second-period goals by Kronwall (16:34), on the power play, and Abdelkader (18:00) for 4-2 advantage.
Datsyuk extended the lead to 5-2 on his second goal of the game, at 11:49 of the third.
Helm added a breakaway goal with 1:50 remaining.
Hedman got the Lightning within 2-1 on his first goal in 32 games at 11:51 of the second. Downie tied it with 5:22 to go in the period.
Cleary put the Red Wings up 1-0 on a deflection of Nicklas Lidstrom's shot during a power play 6:40 into the first. Detroit is 24-2-4 when scoring the game's first goal.27.94653 -82.45927
Pavel Datsyuk scored twice and the Detroit Red Wings beat the Tampa Bay.