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In a closely fought game, it took 28 minutes for the first points to be scored, Luke Dyer going over for Rovers.
The imperious Trent Barrett led the Wigan fightback in the second half, and the former Kangaroo test stand off put them right back in it with a try eight minutes in.
Hull KR stood up to the fight though, and Cooke landed what turned out to be a crucial penalty goal on 55 minutes, stretching Rovers' advantage to 12-6.
Pat Richards added to his earlier conversion by scoring a try of his own with ten minutes left, but he failed to goal that one, and Wigan then failed to score in the final stages, to fall to what ultimately was a disappointing defeat.
It puts Hull KR above their local rivals Hull FC, for 48 hours at least, and moves them clear of bottom placed Salford. Coach Justin Morgan said: "I thought we were the better side, We deserved to win."