Koumas headed his side into a 23rd-minute lead at a third-full Millennium Stadium, only for Robbie Keane to haul Ireland level eight minutes later with the 32nd goal of his international career.
Kevin Doyle then toe-poked the Republic in front on the hour, but when Paul McShane hauled down substitute David Cotterill, it allowed Koumas to score his second from the penalty spot in the 89th minute.
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