Brunner's big game came in his return from a 14-game injury absence as the Devils began a three-game homestand with a lopsided victory.
Cory Schneider earned the victory with 26 saves.
Alexander Steen scored the Blues, who fell for the fourth time in six games.
Brian Elliott started in net but was pulled in the first period after yielding three goals on nine shots.
Jaroslav Halak fared little better, stopping 10-of-14 tries the rest of the way.
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