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.
Rangers backup Steve Valiquette stopped shootout shots by Bryan Little, Slava Kozlov, Kovalchuk, Eric Perrin and Rich Peverley before White flipped the puck past his glove.
Johan Hedberg, who replaced injured starter Kari Lehtonen earlier in the game, denied Markus Naslund, Nik Antropov, Chris Drury, Ryan Callahan, Fredrik Sjostrom and Scott Gomez in the tiebreaker.
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.
The Rangers, 29th in the NHL with their 14.1 percent power-play rate, broke a 1-1 tie with man-advantage goals by Antropov and Drury within a span of 1:30 midway through the period.
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.
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.
Notes: 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.
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