Tootoo and Kostitsyn also had an assist each for the Predators, who are fourth in the Western Conference standings, one point ahead of idle Los Angeles and Phoenix.
Pekka Rinne made 33 saves for the win, which was the eighth in 10 games for Nashville.
Chris Mason started in goal for Atlanta but was lit up for four goals on 13 shots before being lifted in the second period. Ondrej Pavelec stopped nine-of-11 shots the rest of the way.
Kostitsyn scored his second goal at 2:28 of the third period to put Nashville ahead 5-0.
Wheeler broke the shutout at 3:06 on a wrist shot, but Mike Fisher's power-play goal at 6:18 stretched the lead to 6-1.