Patrick Kane had a goal and an assist for the Red Wings (37-35-7), who avoided elimination from Stanley Cup Playoff contention with the win and moved six points behind the Montreal Canadiens for the second wild card from the Eastern Conference with three games remaining for each team. Alex Lyon made 18 saves.
Detroit had lost two in a row, including 4-1 at the Florida Panthers on Thursday.
"They showed each other a lot, not necessarily what they showed me -- it's way more important they play for their teammates," Red Wings coach Todd McLellan said. "We had a good discussion this morning about what we expected from each other and I thought they played to it. Didn't look [it] right off the bat, getting scored on early, but we stayed with it and found a way to scrape out a win in a building that's really tough to play in."
Gage Goncalves scored twice, Anthony Cirelli had a goal and an assist, and Brandon Hagel had two assists for the Lightning (45-26-8), who have lost four of five, with three of those losses coming in overtime or the shootout. Andrei Vasilevskiy made 22 saves.
By earning a point, Tampa Bay clinched a top-three finish in the Atlantic Division, also moving within two points of the Toronto Maple Leafs for first place and two points ahead of third-place Florida. Toronto has played one fewer game.
"Ultimately you want to get the result, you want two points, but what we're doing now with our process has not been right for the last little bit," Lightning coach Jon Cooper said. "It's unfortunate because we've put ourselves in a pretty good situation, a lot of hard work has gone into 70-plus games and now it looks like we're playing preseason games."
Cirelli gave Tampa Bay a 1-0 lead at 1:01 of the first period when he scored on a breakaway set up by a backhand pass up the middle from Hagel.
"This time of year, we've got to go that full 60, that right mentality in order," Hagel said. "We've got a really good hockey team here. We have a lot of belief that we have the team to do it. Every guy here believes that we can do some damage. Right now I'd say it's a little bit frustrating the way we're losing."
Kane tied it 1-1 at 4:07 of the second period, scoring on a one-timer from the left hash marks after Lightning defenseman Victor Hedman knocked the puck off Kasper's stick.