On some forum some years ago, IGN or GAF, the word "solved" was used a lot in a thread similar to this one, as if the game were a problem of some sort.
As for me, "beat" or "finished" is descriptive enough.
Thinking about it, "beat" would seem to indicate the game and I were in some sort of competition against one another. So maybe "finished" is more appropriate? On the other hand, for difficult games (or higher difficulty levels, as the case may be), perhaps "beat" is warranted after all.
Eh, whatever.