Shots were 35-34 in Montreal's favor and Ben Bishop also was solid in goal for the Lightning.
The loss left them 3-5-3 since Jon Cooper replaced Guy Boucher as head coach.
"I haven't been here that long, but that was the best game we've played from start to finish," Cooper said. "I thought we controlled a lot of the play.
"I don't know what the scoring chances were, but I'd have to say we were on the positive end of that. But unfortunately, we're on the negative end of the score. But if you can play consistently the way we did, we're going to win more games than we lose."
Montreal took the lead 4:04 into the second when Galchenyuk was parked at the edge of the crease to slip in a bouncing pass from Brendan Gallagher. It was the rookie's sixth goal in his last eight games.
Gionta then redirected Tomas Plekanec's pass behind Bishop after a turnover at the Tampa Bay blue line at 15:05.
The Lightning took advantage of their first power play when Panik made a deft move to take in Victor Hedman's pass and lift it over Price at 17:36.
Lecavalier tied it 5:07 into the third when Hedman's shot deflected off some skates in front and left a wide opening for the Bolts' captain to score his 10th of the season.
The Lightning took a string of penalties in the third period and the last one cost them when Gionta banged in the winner after Yannick Weber's shot rebounded off the end boards.
"It was a scrambly play and it's nice to capitalize on that, but the best thing is we're getting back to winning ways and getting more confidence," Gionta said. "It's big for our confidence.
"When you lose a few tough games, you lose that little edge, so it's nice to get that back."
Salo left early in the second period with an upper body injury and did not return. Cooper said it did not appear to be serious.
NOTES: Brandon Prust sat out and Michael Ryder returned to the lineup for Montreal. Prust is expected to be back for a home game Saturday against Washington. The Canadiens returned rookie Nathan Beaulieu to AHL Hamilton, as Raphael Diaz is expected to return from a concussion on Saturday. ... Tom Pyatt was scratched for Tampa Bay.