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.
It was the first time that new Lightning general manager Steve Yzerman faced his old team. He played 22 seasons before spending five more as a vice president with Detroit.
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. An octopus — a tradition at Detroit's Joe Louis Arena — was thrown onto the ice after the goal.
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.
Lidstrom had two assists, giving him four in his last 12 games.
Datsyuk made it 2-0 with 5:03 left in the first. The center has eight goals and 16 points in his last 13 games.
Jimmy Howard stopped all six shots he faced, including an in-close backhander by Downie, during a 4-minute Tampa Bay power play midway through the first.
Tampa Bay right wing Martin St. Louis had an assist. He has no goals and three assists in 15 games against Detroit.
Notes: Red Wings senior vice president Jimmy Devellano, one of Yzerman's front-office mentors, and Chicago senior adviser Scotty Bowman, who was Yzerman's coach when Detroit won three Stanley Cups, were at the game. ... NHL commissioner Gary Bettman also was in attendance. ... Detroit D Brad Stuart returned after missing 15 games due to a broken jaw. ... Tampa Bay C Nate Thompson played after sitting out three games with concussion-like symptoms. ... It was just third time in the last eight years that Detroit played the Lightning in Tampa.