Earlier this month on July 16 2016, the San Diego Padres won a game against the San Francisco Giants in the 10th inning when the Giant's pitcher balked. I had never seen this before but doing a quick search I realized this was not the first time a team won by a balk. I'd like to know how many times in MLB history a game has been decided by a balk?
As far back as 1914, the number of times a "walk-off balk" (ie, a balk that ended the game) has taken place is 21(1).
It's possible that the away team can score on a balk to take the lead during the top of the 9th inning (for example), then the home team does not score during the bottom of the 9th inning, securing a victory for the away team. This would fit the criteria of "a game [that] has been decided by a balk" but it is not a "walk-off balk."
Therefore, the minimum number of times a game has been decided by a balk is 21, but there may be more instances in which a game has been decided by a balk but does not fit under the definition of a "walk-off balk."