Trailing 1-0 midway through the first period, Kostitsyn tallied off a feed from Mike Fisher, then found Kline with a pass across the crease 4 1/2 minutes into the second, putting the Preds in front to stay.
Chris Mueller also found the net as Nashville continued its recovery from a four-game skid with a third consecutive victory.
Pekka Rinne turned aside 24 shots to earn the win.
Shawn Horcoff and Corey Potter tallied for Edmonton, which fell for the third straight game.
Devan Dubnyk stopped 23-of-26 shots for the Oilers.