The goalie has never lost to the Lightning, going 15-0 in his career against the defending Stanley Cup champions. He stopped Vinny Prospal on a breakaway with four minutes left in the second.
"I don't know why," Osgood said. "I've just always enjoyed playing in this building and playing against them. It's just one of those things. I can't pinpoint it other than I like playing against them."
Mikael Samuelsson and Datsyuk scored 35 seconds apart early in the third to give Detroit a 5-3 lead. Samuelsson scored from along the right wing boards at 2:12 before Datsyuk picked up his second goal of the game less than a minute later.
"We had some mental collapses in the third," Tampa Bay center Tim Taylor said. "There's no excuse for that. We let a game slip by."
Datsyuk has five goals and 10 points during a five-game point streak.
Brendan Shanahan extended the Detroit advantage to 6-3 with 10:29 left. He has four goals and 13 points during a six-game point streak.
"I thought the game was a lot of fun," Detroit coach Mike Babcock said. "It's zooming out there against those guys. I'd love to play these guys five or six times because I think it improves your play to play against talent like that."
Rob DiMaio, Dave Andreychuk and Ryan Craig scored for Tampa Bay. John Grahame, who has lost two straight after winning a franchise record nine consecutive decisions, was pulled in the third after allowing five goals on 24 shots.
Lightning coach John Tortorella wasn't happy with Grahame's performance.
"I thought we played very well, but do the math," Tortorella said. "Eight scoring chances, five goals. It's frustrating."
DiMaio put the Lightning up 1-0 on a power-play goal in the first. Datsyuk tied it at 1-all from the low slot after making a couple nice moves later in the period.
Tampa Bay went back ahead on Andreychuk's power-play goal in the second. Andreychuk, the all-time NHL power-play goal leader with 273, has scored twice in the last four games after an 18-game goal-scoring drought.
After Franzen scored for Detroit, the Lightning took their third one-goal lead, at 3-2, on Craig's first NHL goal. The Red Wings pulled even again on Woolley's goal late in the second.
"I wasn't able to make the saves tonight," Grahame said. "The third goal (Woolley's) was the one ... we have all the momentum and are playing good. I've got to make that save. Can't let that in."
Woolley's goal was his first in 28 games. He also had two assists.
- Andreychuk moved past Mike Gartner into sole possession of 24th place on the all-time points list with 1,336.
- Tampa Bay LW Ruslan Fedotenko didn't play because of a strained left hip.
- Ryan Craig and RW Darren Reid, who were called up from Springfield of the AHL, made their NHL debuts. They replaced C Martin Cibak and LW Dmitry Afanasenkov, who were scratched.
- Red Wings D Mathieu Schneider left after four shifts with a charley horse.
- Detroit LW Henrik Zetterberg had an assist, giving him at least one point in 16 of 17 road games this year.