The Foals struck on the break with Kone nodding in at the far post after a quick move at the end of a game that had few highlights over 90 minutes.
Wolfsburg had missed two golden chances in second-half stoppage time through Jonas Wind to snatch a late winner.
Bundesliga leaders Bayer Leverkusen take on Paderborn on Wednesday while Borussia Dortmund travel to in-form VfB Stuttgart for a place in the last eight.
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