Pat Maroon scored, and Andrei Vasilevskiy made 27 saves for the Lightning (7-1-1), who swept the two-game set following a 5-1 win here Wednesday.
"Obviously we knew they were going to come out hard and play their best," Tampa Bay defenseman Mikhail Sergachev said. "They are on a long road trip and want to get some points, obviously. We didn't come out with the right approach. We didn't play well and gave them a lot of scoring chances. The second period showed that.
"I felt good about the third period. The boys dialed in and shut them down. [Vasilevskiy] made a huge save there. I hope we can keep playing like that."
Vladislav Namestnikov scored for the Red Wings (2-8-2), who are 0-6-2 in their past eight games and play their next four on the road. Thomas Greiss made 29 saves and is 0-7-2 in his first nine games with Detroit.
"That's more conducive to the type of hockey we have to play to be successful," Detroit coach Jeff Blashill said. "They leaned on us a bit in the third, but up until that point, we hadn't given them much. We're going to be in those 2-1 games and we have to find ways to win them."
Goodrow put the Lightning ahead 2-1 with his first goal of the season at 17:29 of the second period, scoring on a one-timer inside the point off a pass from Jan Rutta.
"I just lost that one," Griess said. "I couldn't really see it."
Goodrow then scored an empty-net goal with 28 seconds remaining for the 3-1 final.
"When it's time to close out a game, even when you're not at your best, you've got to find a way to win some of those," Lightning coach Jon Cooper said. "We did against a very determined Detroit team."