A pebbling game on a simple graph consists of moves that remove two pebbles from a vertex of the graph and add one pebble to an adjacent vertex. We consider an impartial two-player pebbling game, where the winner of the game is the last player to make an allowable pebbling move. In this paper, we characterize the winning positions when this game is played on a complete graph K_n with at least n + 2 pebbles if n ≥ 5 is odd and at least n + 7 pebbles if n ≥ 6 is even. This characterization is independent of how the pebbles are initially distributed on the vertices and depends only on the parity of the total number of pebbles used in the game.