Kozlov set up Shawn McEachern's go-ahead goal with 7:08 left in the third period, then scored with 2:01 remaining to reach the 20-goal mark for the eighth time in his career.
Ilya Kovalchuk's empty-netter capped the scoring with 55 seconds left. It was his 37th goal, tying him with Columbus' Rick Nash for the league lead.
Dany Heatley recorded three assists for the Thrashers, who are 10 points behind the New York Islanders for the final Eastern Conference playoff spot.
Celebrity Families of 2014 [PHOTOS]