The win was Tampa's sixth straight road victory after starting off the season 3-14-3 away from the St. Pete Times Forum.
They will aim to complete a sweep of their three-game road trip Saturday in Atlanta.
On the game-winner, Brad Richards fired a slap shot from the top of the right faceoff circle that was stopped by Nashville goaltender Dan Ellis. Prospal was standing just to the right of Ellis, and was able to send the rebound between the goaltender's pads for his 23rd goal of the season.
"We turned the puck over, got it back, and got a shot on the net," Prospal said. "We were able to get the rebound because we were hungrier than they were, and we were luckier than they were."
"It hit my chest and went down," Ellis said of Richards' shot. "It was a fluttering type of shot that I needed to control, and [Prospal] was able to pounce on it before I was."
Alexander Radulov broke a scoreless tie at 12:37 of the first period, scoring his 21st goal with a wrist shot from the left faceoff circle that sailed over the glove of Tampa Bay goaltender Johan Holmqvist.
St. Louis' deflection of a Lecavalier pass was stopped by Ellis, but the rebound ended up to the left of the net. Richards skated in and poked the puck past a diving Ellis.
"We are still in it, there is a lot of hockey left," Richards said. "We have to get it in our heads that we haven't accomplished anything yet. The road has been good. We have to put some more together."
The Predators were 0-for-2 on the power play. They have scored just one power-play goal in their past six games, going 1-for-21 over the stretch.
"I didn't think we brought our best effort tonight, but I don't want to take anything away from Tampa Bay," Nashville coach Barry Trotz said. "They had a lot of urgency in their game and they found a way to take it into overtime."
Nashville lost its first game against a Southeast Division team after winning its first three such games. Tampa Bay evened its record against Central Division opponents at 2-2-0.
Tampa Bay entered the game with the sixth-best power play in the NHL, but was 0-for-1 in the game. Tampa spent just 49 seconds on its one power play before it was negated by a Dan Boyle hooking penalty that broke up a potential short-handed breakaway by Nashville's Martin Gelinas.
"We know where we are at," Lightning coach John Tortorella said. "We've won a couple of road games here, and with our division the way it is, it gives us a chance to stay in the hunt."
Nashville center Vernon Fiddler appeared in his 200th career NHL game, all played for the Predators. ... Tampa Bay was playing its first road game in Nashville since Feb. 5, 2004 ... Nashville forward Martin Erat left the game in the second period with an upper-body injury that Trotz said was a twisted back and not believed to be serious.