Drury scored his fifth of the season with 27.5 seconds left.
Martin St. Louis had the other goal for the Lightning, who lost for the second straight night after a three-game winning streak.
They are 4-2-4 since a pair of 2-1 losses to the Rangers during the season-opening series in the Czech Republic.
Henrik Lundqvist made 36 saves for the Rangers, who snapped their first two-game skid of the season. Mike Smith stopped 30 shots for Tampa Bay.
The Rangers' much-maligned power play had failed on all six chances in the two previous contests. In New York's 2-1 loss to the Islanders on Tuesday, the Rangers allowed both goals while on their power play and have given up an NHL-high five short-handed goals.
Nearly 90 seconds into their first chance, Nikolai Zherdev raced down right wing and fed a perfect pass into the slot to Callahan, who received it while crouching and fired a one-timer past Smith to make it 1-0 with 3:51 left in the first.
The Lightning didn't get a power-play opportunity until Zherdev was caught cross-checking 42-year-old agitator Gary Roberts in front of the Lightning bench 13:13 into the second. Before the retaliation, Zherdev had been roughed up by Roberts, Chris Gratton and Adam Hall.
But Drury pushed the Rangers' lead to 2-0 with New York's first short-handed goal of the season. He picked off a pass from Lightning defenseman Paul Ranger in the neutral zone and took off on a breakaway.
Smith dropped to his side in front of the oncharging Drury, who deftly moved to his right to get around the fallen goalie, and scored into an empty net with 5:37 left in the second.
St. Louis got the Lightning to 2-1 with 1:24 remaining, scoring his second goal in two nights. He had a goal against the Rangers in the opening game of the season and didn't net another until the Lightning's 4-3 shootout loss at New Jersey on Wednesday.
Drury restored the Rangers' two-goal lead 35 seconds into the third with his power-play tally.
Malone made it 3-2 at 7:58, but Staal responded at 8:27.
Notes: The Lightning activated RW Radim Vrbata from the injured list after he missed five games due to a groin injury. ... The teams will meet once more, Nov. 26 at Tampa Bay. ... The Rangers were outshot 15-9 in the second period after holding a 12-9 edge in the first. ... Former New York D Marek Malik heard boos every time he touched the puck on his first trip back to Madison Square Garden. ... Smith received nine penalty minutes with 1:41 left after a scrap with Aaron Voros.
