The teams combined to miss the first 10 shots of the shootout before St. Louis bested Jonas Gustavsson. Ben Bishop then turned aside Johan Franzen to clinch the victory and end Tampa Bay's two-game skid.
Detroit has lost three in a row and slipped behind Tampa Bay into fourth place in the Atlantic Division.
Defenseman Kyle Quincey scored a rare goal for Detroit in the opening period, his first of the year and his 20th in 326 NHL games.
Nikita Kucherov tied the game for the Lightning with 44 seconds to go in the middle period.
Bishop finished with 28 saves and Gustavsson stopped 26 shots.