Maribor held out until the 82nd minute but Bayer Leverkusen - who top the Bundesliga this week - refused to be denied as they saw late pressure bring two goals.
A close range header from Boris Zivkovic set the German side on their way and veteran striker Ulf Kirsten drove in a fierce shot as the game went into injury time to seal the win when he threw defenders off balance with quick foot work.
Maribor mounted a stern defence but saw their best chance wasted when Kliton Bozgo fluffed a clear run on goal.
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