"We had the lead. I think that was the key and we jumped on them right away," Jagr said. "We scored a quick two goals and that was the difference."
Lundqvist stopped 14 shots in earning his seventh shutout and 14th in three NHL seasons.
Kari Lehtonen made 35 saves but failed to win for the fourth straight outing -- all losses for the Thrashers, who are three points behind Southeast Division-leading Carolina.
"Everybody seemed sleepy or tired," Thrashers coach Don Waddell said. "We've had slow starts a couple of games in a row and it's killing us."
The Rangers came out with changes to their top three lines, and the moves paid off.
Avery, playing with Jagr and Scott Gomez, got New York on the board 12:12 in. He found a loose puck in the slot and shoved it past Lehtonen for his fourth of the season and second in four games since returning from his latest wrist injury.
Jagr earned his second assist 3:20 later, sending Dubinsky into the zone with a pass from inside the blue line. Dubinsky carried on the right side and eluded falling defenseman Alexei Zhitnik before scoring his seventh.
New York showed jump and life, elements that were sorely lacking in a home-and-home sweep by Boston last weekend and during a 2-6-2 slide. The Rangers are in ninth place in the Eastern Conference, one point behind Boston and the postseason cutoff and three points ahead of Atlanta.
"I am glad we won, but we have a long way to go," Jagr said. "We have to win a lot more games to get in the playoffs."
The shot total was 14-0 when Dubinsky scored, and Atlanta didn't get its first puck on net until Bobby Holik's shot with 3:14 left in the period.
"We looked like a junior team. It's not the way it's supposed to be," Lehtonen said.
New York had a decided edge in shots early in Sunday's 3-1 loss at home to the Bruins, but that hardly impressed Renney because the Rangers had very few scoring chances. That changed against the Thrashers.
"You're not going to score unless you shoot the puck," Renney said. "We didn't give them a whole lot to work with. We are still trying to get ourselves corrected here.
"We understand that they are an explosive team, so we thought that we would have our hands full. We got a good start, a good jump on them. We were a pretty relentless team," he said.
Renney ripped into the Rangers following Sunday's loss and threatened changes. The only difference in the active roster was defenseman Marek Malik came back in for Paul Mara, who was benched.
Forward Marcel Hossa was scratched for the fourth straight game, missing an opportunity to play against brother Marian of the Thrashers.
Marian Hossa became Atlanta's biggest offensive threat during the final 40 minutes after leading scorer Ilya Kovalchuk was ejected with 35.7 seconds left in the first period when he drove Rangers defenseman Michal Rozsival into the boards from behind after elbowing him earlier.
Rozsival was cut and stayed down for a few minutes as Colton Orr came to his defense. The Rangers got a three-minute power play but couldn't take advantage. Rozsival wasn't seriously hurt and returned.
"It's a contact sport," Kovalchuk said. "I wasn't trying to kill anybody."
Straka scored a shorthanded breakaway goal 8:36 into the second after Atlanta's Slava Kozlov went off for a line change, allowing Straka to chase down a loose puck and skate in alone on Lehtonen.
Jagr completed the scoring with his 15th goal at 4:03 of the third. On Thursday, New York will try to finish off the rare back-to-back homestand against the same opponent.
"They were missing Kovalchuk, and he is a big part," Jagr said. "Next game they are going to be better."
The Rangers will retire the No. 2 in honor of former defenseman Brian Leetch before playing the Thrashers on Thursday . ... Kovalchuk, who leads Atlanta with 37 goals and 63 points, hasn't scored in five games.
Copyright by STATS LLC and The Associated Press
Arena: Madison Square Garden Location: New York, New York
Referees: Kerry Fraser Linesmen: Scott Driscoll, Anthony Sericolo