The hosts improved following the early scare, and they were rewarded for an extended spell of pressure midway through the first half when Leverkusen stopper Bernd Leno raced off his line to bring down Marcell Jansen.
Returning skipper Van der Vaart stepped up give his side the advantage with a calm finish from the penalty spot.
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Despite dominating possession after falling behind, Leverkusen were finding it difficult to trouble Drobny as the home defence limited the opportunities on offer.
Chances were rare for both sides in the second half, but Heiko Westermann would have doubled his side's advantage on the hour mark had he managed to keep his header from sailing over the crossbar.