Wright was selected by the Los Angeles Dodgers in the 7th round of the 2003 Major League Baseball Draft and signed with them on June 10 2003.
He made his professional debut with the Ogden Raptors in the Pioneer League in 2004 and played for the Columbus Catfish and Vero Beach Dodgers in 2005. With Columbus, he pitched in 30 games and accumulated a 1.93 era as a relief pitcher.
In 2006, he pitched in 25 games with Vero Beach, acquiring a 1.48 era and was promoted to the Jacksonville Suns, where he began 2007 before a mid-season promotion to the Las Vegas 51s.
In December 2007, he was selected by the Houston Astros in the major league portion of rule 5 draft from the Los Angeles Dodgers. Wright made the Astros 2008 opening day roster as the only left-handed pitcher in the bullpen. On March 31, 2008, Wright made his major league debut against the San Diego Padres and pitched 1/3 of a scoreless inning. Wright collected his first major league win on April 4, 2008, against the Chicago Cubs after pitching 1/3 of a scoreless inning again. He collected his first major league save on July 29, 2008, against the Cincinnati Reds coming in during 9th inning with two outs, and recorded the final out of the game for the save.
On July 16, 2009, Wright was called in to pitch in a game against the Cubs in the second inning after starter Roy Oswalt was forced from the game by a strained back. Wright batted in the top of the third inning and singled to left field for his first Major League hit. Later that night, it was reported that he was rushed to the hospital after showing signs of appendicitis. He was diagnosed with dehydration.
On August 12, 2013 he was traded to the Tampa Bay Rays for cash considerations. With Tampa Bay, Wright made his playoff debut in 2013, appearing in two games of the Rays' ALDS matchup with the Boston Red Sox. After the season, Wright was non-tendered by Tampa Bay, making him a free agent.
On December 4, 2013 he agreed to a one-year, $1.425 million deal to pitch for the Chicago Cubs. The contract became official on December 16, 2013. He was non-tendered by the Cubs on December 2, 2014.
On December 16, 2014 he agreed to a one-year deal to pitch for the Baltimore Orioles. Wright was designated for assignment on July 15, 2015.
On July 29, 2015, Wright signed a minor league deal with the Los Angeles Angels of Anaheim.
Wright joined the Arizona Diamondbacks for 2016 spring training, and was released on March 28, 2016. On April 12 he signed a minor league deal with the Boston Red Sox and was assigned to Triple-A Pawtucket Red Sox, where he appeared in 19 games and was released on July 11.
On January 27, 2017, Wright signed a minor league contract with the Texas Rangers. He was released on July 23, 2017.