Magnus Paajarvi had the other goal for the Blues, who have won all four of their games against Nashville this season.
Ryan Miller stopped 20 shots to win his third game in as many starts since being acquired from the Buffalo Sabres on Friday. Nick Spaling scored for the Predators, losers of six of seven.
Cole scored the game's first goal at 4:03 of the opening period after Steve Ott won a faceoff in the left circle and Cole beat Nashville goaltender Pekka Rinne with a wrist shot high to the far side.
Ott's assist was his first point as a Blue.
Nashville drew even at 10:28 of the first on Spaling's 11th goal of the season.