The Thrashers, who are now playing only for pride, rallied from a three-goal deficit behind Kovalchuk. Todd White then scored the only goal in the shootout to give Atlanta a 5-4 win over the Rangers on Thursday.
The Rangers led 4-1 late in the second period.
"Where is our killer instinct when we're playing someone lower in the standings?" asked Rangers backup Steve Valiquette, who stopped shootout shots by Bryan Little, Slava Kozlov, Kovalchuk, Eric Perrin and Rich Peverley before White flipped the puck past his glove.
After the Rangers built the big lead, Kovalchuk had a hand in each of the next three goals.
Kovalchuk scored late in the second period and assisted on third-period goals by Colby Armstrong and White as Atlanta tied it with 6:50 remaining in regulation.
The Rangers are one point behind sixth-place Pittsburgh in the Eastern Conference playoff race. New York will travel to face the Penguins on Saturday.
"We have to stay up, of course, because we have a big game coming up in a couple of days," Valiquette said. "Nobody wants to get too down and too negative, but tonight wasn't exactly a forward performance for us."
The Thrashers, already eliminated from playoff contention, have won eight of 11.
"We need to play for our fans right now," Kovalchuk said. "We want to be better and make sure the fans have a good time."
Atlanta improved to 7-1 in shootouts.
"To hang in there and hang in there and hang around and not quit in the third ... we haven't got a lot to play for, but I think we owe our fans and even ourselves a solid effort no matter what happens," Atlanta coach John Anderson said. "And that's what happened. We went out and played as hard as we could and got a couple of goals."
"I think we are fortunate enough to get a point, and we were very fortunate to get that," Rangers coach John Tortorella said.
Kovalchuk, chosen as Thrashers captain this season, scored with 20.6 seconds remaining in the second period to start the comeback.
"To give one up on the road with under a minute left in the period is a killer," said Rangers captain Chris Drury, who had a goal and two assists. "That's not why we lost, but it certainly changed things in their locker room going into the third."
The Rangers appeared on their way to a win when they took advantage of long-awaited success on the power play and grabbed the lead in the second period.
New York, 29th in the NHL with a 14.1 percent power-play success rate, broke a 1-1 tie with man-advantage goals by Antropov and Drury within a span of 1:30 midway through the period. Naslund added another second-period goal for the 4-1 lead.
Drury's shot from the right wing came on Valiquette's first career assist. Drury added two assists, and Nikolai Zherdev also scored for the Rangers.
Valiquette stopped 26 shots, but couldn't hold the big lead. Kozlov tied it 1-1 for the Thrashers early in the second period.
Lehtonen sustained an undisclosed upper body injury and left after the first period. Hedberg made 14 saves before his perfect shootout performance.
"It's been a rough year for us, but later in this season things have been going in the right direction," Hedberg said. "Going into next year, we want to have some momentum."
Drury had a first-period goal disallowed following a long video review that determined that his stick was above the crossbar when he knocked the puck past Lehtonen.
- New York managed only four shots in the third period.
- Atlanta C Jim Slater returned after missing five games due to head and shoulder injuries sustained March 14 at Buffalo.
- The Rangers have seven games left, all against playoff-bound teams.
- Lehtonen stopped nine of 10 shots.
- Kovalchuk, awarded assists on goals by Armstrong and White, has nine goals and 13 assists in 12 games.