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Starting this previous season this rule only applies to post season playoff games that go into overtime. Previously in overtime, the first team to score either a field goal or a touchdown would win the game. Now, the rule is that if the team that wins the coin toss and elects to receive scores only a field goal on their first drive, the other team will still get the ball with the chance to score as well.
So if the Cowboys win the toss against the Giants and choose to receive(of course they would), they must score a touchdown on their first drive in overtime to win the game. If the Cowboys only manage to kick a field goal on their first drive, they'll have to kick off to the Giants afterward. Now, if the Giants manage to score a touchdown they will win the game. If the Giants only kick a field goal to tie the game with the Cowboys again, they must kick off to the Cowboy and the game becomes sudden death. If the Cowboys managed to score a field goal on their first possession of Overtime but the Giants don't score at all during theirs then the Cowboys win.
Also, at any point during the Cowboy's first possession of Overtime the Giants can win by running a turnover in for a touchdown or by scoring a safety.
It's funny, I honestly thought I could answer this one really simply from what I already knew, yet I still needed to consult four different websites to figure everything out. The best one with all the answers is here: