The Lions, who showed very little in attack, led 12-9 at half time.
But the crucial moment came with just five minutes left, when Cooper crossed for the game's only try.
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Christian Leali'ifano scored the conversion and 3 penalties, while Leigh Halfpenny scored all of the Lions' points, but missed a last gasp kick from halfway to win the game and the series.
The decider will take place in Sydney next week.
