You need to assess why white made the mistake in the first place. Maybe black made a threat as a ploy for white to make mistake. If black did, then black has indeed outplayed white. It also may depend where the mistake was made. If the mistake was made in the opening because black went out of book and wins because of it, then black, again, has indeed outplayed white.
White does have the slight advantage in the beginning because of the first move rule, and with perfect play from both sides, white should win. However, if white does not take advantage of his first move and allows black to equalise and win, then black, again has outplayed white. Consider how many games of chess has been won by mistakes made by either black or white. The game of chess is, in actuality, a game where each opponent tries to incite the other to make a mistake. If black makes a mistake in trying to take advantage of whites mistake, and white goes on to win because of it, then , has white not outplayed black? of course he has! Therefore, if white makes a mistake and black takes advantage of it correctly thus winning the game, then black, again, has outplayed white.
We all make mistakes!!!
There is nothing more satisfying than a clean white piece victory. The thing is that since white has an advantage in the opening-a black piece victory means your oponent made a mistake so you didin't really outplay him. Winning with white is just that much cooler. It's like saying I have the advantage and I pushed it to a win! Makes you feel superior somehow.