We introduce and study balanced online graph avoidance games on the random graph process. The game is played by a player we call Painter. Edges of the complete graph with n vertices are revealed two at a time in a random order. In each move, Painter immediately and irrevocably decides on a balanced coloring of the new edge pair: either the first edge is colored red and the second one blue or vice versa. His goal is to avoid a monochromatic copy of a given fixed graph H in both colors for as long as possible. The game ends as soon as the first monochromatic copy of H has appeared. We show that the duration of the game is determined by a threshold function mH = mH(n). More precisely, Painter will asymptotically almost surely (a.a.s.) lose the game after m = ω(mH) edge pairs in the process. On the other hand, there is an essentially optimal strategy, that is, if the game lasts for m = o(mH) moves, then Painter will a.a.s. successfully avoid monochromatic copies of H using this strategy. Our attempt is to determine the threshold function for certain graph-theoretic structures, e.g., cycles.