Kari Lehtonen was perfect in goal during the shootout after making 30 saves in regulation.
Kozlov lifted the puck over prone goaltender Rick DiPietro and Kovalchuk then sealed the win by beating DiPietro between the pads.
Kovalchuk also scored during regulation, while Marian Hossa added a goal and an assist. Atlanta has won two of its last three.
Atlanta overcame a 1-0 deficit by scoring three goals in an eight-minute span of the second period -- two of them shorthanded.