The 1994 Shell Caribbean Cup had the odd twist that goals scored in overtime counted double for purposes of the tournament. I'm not sure what the motivation for that was, but it had an unexpected consequence. Here's the wiki description of the Grenada vs Barbados game:
Grenada went into the match with a superior goal difference, meaning that Barbados needed to win by two goals to progress to the finals. The trouble was caused by two things. First, unlike most group stages in football competitions, the organizers had deemed that all games must have a winner. All games drawn over 90 minutes would go to sudden death extra time. Secondly and most importantly, there was an unusual rule which stated that in the event of a game going to sudden death extra time the goal would count double, meaning that the winner would be awarded a two goal victory.Here's a grainy pre-HD video exerpt.
Barbados was leading 2-0 until the 83rd minute, when Grenada scored, making it 2-1. Approaching the dying moments, the Barbadians realized they had no chance of scoring past Grenada's mass defense, so they deliberately scored an own goal to tie the game at 2-2. This would send the game into extra time and give them another half hour to break down the defense. The Grenadians realized what was happening and attempted to score an own goal as well, which would put Barbados back in front by one goal and would eliminate Barbados from the competition.
However, the Barbados players started defending their opposition's goal to prevent them from doing this, and during the game's last five minutes, the fans were treated to the incredible sight of Grenada trying to score in either goal. Barbados also defended both ends of the pitch, and held off Grenada for the final five minutes, sending the game into extra time. In extra time, Barbados notched the game-winner, and, according to the rules, was awarded a 4-2 victory, which put them through to the next round.