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Goode and James O'Connor each kicked two penalties, before the former converted a penalty try to put Wasps a 13-6 ahead on 36 minutes.
With O'Connor in the sin bin for a knock on, Shane Geraghty was temporarily put on kicking duty.
Geraghty rose to the challenge to send a penalty between the posts to reduce his side's deficit to four points just before half time.
O'Connor returned to the action and he and Goode again traded penalties.
The Wasps fly-half then completed the scoring by nailing a nerve-settling drop goal.