Barrett grew up in North Tonawanda, New York and Mountain View, California. He earned a degree in kinesiology at Cal State-Hayward in 1988. In college, he was captain of the football team. Prior to pursuing umpiring, he was an amateur boxer. Played high school basketball for Vance Walberg, inventor of the dribble drive offense.
His professional umpiring career began after he attended the Joe Brinkman Umpire School in 1989, and he worked his way up to the Pacific Coast League for the 1993 season. He made his major league debut in 1994. For the next five seasons, Barrett served as a fill-in umpire for vacationing or injured major league umpires. Barrett was one of the 25 umpires promoted in the wake of the Major League Umpires Association's mass-resignation strategy in July 1999. His five years of experience made him one of the most experienced of the 25 umpires called up to fill the sudden vacancies.
He has worked 8 Division Series in (2000, 2001, 2002, 2003, 2006, 2007, 2011, 2014); eight Championship Series (2005, 2008, 2009, 2010, 2012, 2013, 2015, 2016); and three World Series (2007, 2011, 2014).
He was promoted to the position of crew chief prior to the 2013 season.
He was the home plate umpire for David Cone's perfect game for the New York Yankees against the Montreal Expos on July 18, 1999, the first interleague no-hitter in the regular season.
Barrett was behind the plate on August 7, 2004 for Greg Maddux's 300th win.
During a game between the Atlanta Braves and the San Francisco Giants on August 14, 2007, Barrett ejected Atlanta manager Bobby Cox for arguing balls and strikes. The ejection was the 132nd of Cox's managerial career, breaking the major league record previously held by John McGraw.
Barrett was behind home plate at Progressive Field when Los Angeles Angels pitcher Ervin Santana threw a no-hitter against the Cleveland Indians on July 27, 2011.
He was at third base on April 21, 2012 for Philip Humber's perfect game.
Barrett was at first base for a combined no-hitter thrown by six Seattle Mariners pitchers against the Los Angeles Dodgers on June 8, 2012.
He became the only umpire in major league history to be behind the plate for two perfect games on June 13, 2012, when Matt Cain pitched a perfect game for the San Francisco Giants against the Houston Astros.
Barrett was chosen as one of the umpires for the one-game Wild Card playoff between the Baltimore Orioles and the Texas Rangers on October 5, 2012. He also worked the 2013, 2015 and 2016 AL wild card playoff games.
Barrett was the home plate umpire for game 6 of the 2016 National League Championship Series in which the Chicago Cubs won their first National League pennant since 1945.
He is also heavily involved with Christian ministry, and has earned a master's degree in Biblical Studies from Trinity College and Seminary. In 2013, he earned his PhD in theology with an emphasis in Pastoral Ministry from Trinity Theological Seminary.