Since 1988 (25 year span) The AL has won the WS 14 times while the NL has won it 11 times. Easily found many places, including
Wiki
Year, Winner, Loser, Series
2013, Boston Red Sox, St. Louis Cardinals, 4-2
2012, San Francisco Giants, Detroit Tigers, 4-0
2011, St. Louis Cardinals, Texas Rangers, 4-3
2010, San Francisco Giants, Texas Rangers, 4-1
2009, New York Yankees, Philadelphia Phillies, 4-2
2008, Philadelphia Phillies, Tampa Bay Rays, 4-1
2007, Boston Red Sox, Colorado Rockies, 4-0
2006, St. Louis Cardinals, Detroit Tigers, 4-1
2005, Chicago White Sox, Houston Astros, 4-0
2004, Boston Red Sox, St. Louis Cardinals, 4-0
2003, Florida Marlins, New York Yankees, 4-2
2002, Anaheim Angels, San Francisco Giants, 4-3
2001, Arizona Diamondbacks, New York Yankees, 4-3
2000, New York Yankees, New York Mets, 4-1
1999, New York Yankees, Atlanta Braves, 4-0
1998, New York Yankees, San Diego Padres, 4-0
1997, Florida Marlins, Cleveland Indians, 4-3
1996, New York Yankees, Atlanta Braves, 4-2
1995, Atlanta Braves, Cleveland Indians, 4-2
1993, Toronto Blue Jays, Philadelphia Phillies, 4-2
1992, Toronto Blue Jays, Atlanta Braves, 4-2
1991, Minnesota Twins, Atlanta Braves, 4-3
1990, Cincinnati Reds, Oakland Athletics, 4-0
1989, Oakland Athletics, San Francisco Giants, 4-0
1988, Los Angeles Dodgers, Oakland Athletics, 4-1
Also, from Baseball Reference regarding Interleague play this season-
http://www.baseball-reference.com/
From MLB.com-
http://mlb.mlb.com/mlb/history/interleague/records.jsp
Interleague Play Records
Notice that outside of 1997, 1999, 2002 and 2003, the AL has beaten the NL.
There is very little to no evidence that supports your assertion that NL has the advantage over the AL in Interleague play.