Here is a quick fact.
Boston - Fenway Park - 1st Game - The Red Sox defeated the New York Highlanders — later known as the Yankees on April 20th, 1912*
New York - Yankee Stadium - 1st Game - ready for its inaugural game on April 18, 1923 vs. the Boston Red Sox. Yankees Won 4-1
* (The first official game played in Fenway actually occurred on April 9 when the Sox beat Harvard University, 2-0.)