Ryan Callahan and Marc Staal also found the back of the net for New York, which snapped a two-game skid. Nikolai Zherdev and Michal Rozsival each added two assists while Henrik Lundqvist made 36 saves to earn the win.
"It's sort of as advertised with Chris (Drury), this is what he does," Rangers head coach Tom Renney said. "He is our leader and our captain, and he has to demonstrate leadership beyond scoring."
Ryan Malone and Martin St. Louis lit the lamp for Tampa Bay, which had a three-game winning streak snapped Wednesday in New Jersey. Mike Smith allowed five goals on 35 shots.
Starting the third period on the power play, which began near the end of the middle stanza, Scott Gomez dropped the puck back to Drury at the right circle and he fired a one-timer past Smith for his second goal of the night, giving New York a 3-1 lead just 35 seconds in.
Malone cut the deficit back to one with a shot under the arm of Lundqvist to the stick-side after taking a cross-ice pas from Mark Recchi but Staal's marker at 8:27 -- only 29 seconds after Malone's tally -- restored New York's two-goal edge and Lundqvist did the rest.
Prospal found himself alone in front just moments later and for a point-blank chance but couldn't control the puck before being taken out by a defender. Steven Stamkos, the league's latest number one overall pick, was then foiled on a breakaway chance.
The Lightning took a pair of minors with 3:25 remaining, giving the Rangers a 5-on-3, and after some rough stuff extended the advantage, Drury capped a big night by completing the trick with just 28 seconds left to play.
"In the last two periods, we outchanced them outshot them, and did a lot of good things," Lightning head coach Barry Melrose said. "Our team did a lot of good things. We battled hard."
New York took advantage of its second power-play opportunity in the opening frame. Zherdev carried the puck into the zone along the right boards and fired a pass through the slot to Callahan whose one-timer found twine.
The Rangers made it a 2-0 contest with a short-handed marker with 5:37 remaining in the second period. Drury picked off an errant pass from the Lightning defense at the blue line and skated in uncontested. Drury faked Smith to the ice, put on the brakes, and scored easily into an open net.
"When you're struggling to step up and you're not clicking -- and then you have an odd man rush -- you have to take advantage of it," Drury said.
Tampa Bay got on the board with 1:24 left in the second when St. Louis took a pass from Vaclav Prospal just inside the blue line and with a defender draped on his back, lifted a backhand shot over the shoulder of Lundqvist.