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Tahiti, boosted by their French club-based players, beat the Cooks 13-9 in Rarotonga last August to win the tournament for nations in the region ranked below Fiji, Samoa and Tonga.
World Rugby said in a statement on Friday the Cook Islands had asked for an investigation into the eligibility of players in the Tahitian side.
"Independent Judicial Officer Tim Gresson (New Zealand) heard the case and, having considered all the facts, ... he determined that the Tahiti Rugby Union had breached Regulation 8 by fielding two ineligible players for the match," the world governing body said.
"In his determination, the Judicial Officer decided, in light of the clear World Rugby Regulation 8 breach, that the result be overturned and directed that Cook Islands be confirmed as the winner of the match."
The Cook Islands will now face the winner of the Asian Rugby Championship, involving Malaysia, Hong Kong and South Korea, in a playoff to enter the repechage competition that will consist of four teams.
World Rugby were embroiled in another qualifying controversy this week with Spain asking them to replay their match with Belgium after an 18-10 loss in Brussels resulted in Romania advancing to next year's tournament in Japan.
The match was controlled by Romanian referee Vlad Iordachescu, who was appointed by Rugby Europe.
Iordachescu was pursued by furious Spain players and several irate supporters and needed to be escorted off the pitch by competition officials.