Gagne scored two goals, the second into an empty net in the final minute, to help the Lightning beat Carolina 4-2 and avenge a home loss to the Hurricanes on Friday night.
The result broke Carolina's three-game winning streak and the Lightning's four-game winless streak, dropping the Hurricanes five points behind eighth-place Buffalo with seven games left.
Gagne has 16 goals this season, six in five games against Carolina.
"We knew it was going to be a tight game," Gagne said. "To get this one was huge for us. We battled really hard as a team from goalie to forward to defenseman. It was really a playoff-type game, and it's good to come here on the road and beat a team that's very desperate for points."
"The reality is we didn't work last night," Tampa Bay coach Guy Boucher said. "We decided today we were going to work, and that's how we played all year long. This is exactly how we won our game. The minute we don't work we lose against anybody."
"Now we're going to need help," Hurricanes coach Paul Maurice said. "We're going to have to come pretty close to running the table from here on out."
Staal gave Carolina the lead on a power play at 7:07 of the first, receiving Stillman's pass at the spot of the right circle and sliding it outside Smith's right skate for his 31st goal of the season.
St. Louis tied it for the Lightning on a rush at 13:56, taking a pass from Lecavalier and beating Ward stick-side for his 27th of the season.
"This was good," Lecavalier said. "We played 60 minutes, played hard, played physical and did a lot of good things. Now we have to carry it over to the next game."
Gagne made it 2-1 with 2:24 left when he stole the puck from Joe Corvo just inside the Hurricanes' blue line and skated in to beat Ward.
Cole tied it with 50 seconds left in the period, taking Stillman's pass in traffic on the right doorstep and beating Smith for his 23rd of the season.
But Moore put the Lightning ahead again on the power play with 2:07 remaining in the second, beating Ward off an assist from Lecavalier.
Hurricanes D Jamie McBain was back in the lineup after missing six games (shoulder). He assisted on Staal's first-period goal. ... Carolina D Joni Pitkanen missed the game because of illness. ... Lightning D Randy Jones (lower body) missed his ninth straight game.