Montreal has won four consecutive games, including three on the road, and the sixth-place Canadiens are only two points behind Tampa Bay in the East.
The Southeast Division-leading Lightning have lost three in a row and are just one point ahead of Washington.
Price stopped all but three of the 116 shots he faced during a three-game road sweep of Southeast Division opponents Atlanta, Florida and Tampa Bay.
"We all bought into this road trip," he said. "We made some stops and scored some goals and came out on top, so it was a great road trip for us."
"It didn't surprise me, it shocked me," Lightning coach Guy Boucher said of his team's slow start. "Everything we do now falls on everybody, whether we've got two guys that are out there asleep or whether we've got one or 15."
Gill, a defenseman, has scored goals in back-to-back games after going 99 without one.
"It's always fun to score, and I'd love to do more of it, but it's not like this is the first time," he said.
Dominic Moore, who set up the other Lightning goal by Vincent Lecavalier, made it 2-1 during a power play 1:25 into the second period. It was the first time during their winning streak that the Canadiens failed to kill a penalty.
Pacioretty restored Montreal's two-goal advantage on a power play with 5:55 left in the second, then made it 4-1 at 1:18 of the third.
"Our two power plays were huge," Canadiens coach Jacques Martin said. "I think that two goals from him were huge, and a huge factor in the game. He really stepped up."
Lecavalier closed the scoring with 5:53 to go, but Tampa Bay netted fewer than three goals for the seventh time in eight games.
"It's definitely a struggle right now," he said. "We've got to get back to taking the shots and putting back rebounds. We just couldn't get it done."
Brian Gionta, the Montreal captain, had an assist in his 600th NHL game. Then he praised Price.
"He is playing great out there," Gionta said. "He's moving his feet and making the plays. When he's doing that, he's his best."
- Tampa Bay C Steven Stamkos, the league's leading scorer with 41 goals, has only one in 11 games.
- Montreal LW Andrei Kostitsyn, with an assist on Desharnais' goal, has eight points in six games.
- The Lightning are nearing the end of a 19-game stretch that included only three road games.
- The Canadiens improved to 24-1-4 when leading after two periods.