Hainsey's snap shot past Jean-Sebastian Giguere completed a dramatic rally for the Thrashers, who trailed 2-0 entering the third period.
Andrew Ladd's power-play goal 23 seconds into the frame pulled Atlanta to within a goal, and with less than 5 minutes remaining in regulation Tim Stapleton fired home the rebound of a Zach Bogosian shot to tie it.
Chris Mason stopped 22-of-24 shots as Atlanta improved to 2-7-2 in February.
Nikolai Kulemin and Phil Kessel scored for the Maple Leafs.
Starting Toronto goalie James Reimer stopped all 18 shots he faced before being forced to leave the game early in the second period after being struck in the head by the knee of Thrashers forward Evander Kane.
Giguere entered and surrendered all three Atlanta goals on 25 shots.
2014: The Year in Music [PHOTOS]