Sosa was signed as an amateur free agent by the San Francisco Giants on April 23, 2004. He began his professional career in 2006 with the AZL Giants. With them, he went 2-1 with a 3.90 ERA in nine games. He also struck out 41 batters in 32 innings. In 2007, he played for the San Jose Giants and Augusta Greenjackets. He went 6-0 with a 0.73 ERA in 13 games with the Greenjackets and 5-5 with a 4.38 ERA in 14 starts with the Giants. He went a combined 11-5 with a 2.58 ERA in 27 games. In 125 innings, he struck out 139 batters. He also played for the Estrellas de Oriente of the Dominican Winter League that year. For the San Jose Giants and Augusta Greenjackets in 2008, he went a combined 3-4 with a 4.21 ERA in 57 innings. He played only two games with Augusta.
Houston Astros
On July 19, 2011 Sosa was traded to the Houston Astros along with Jason Stoffel for infielder Jeff Keppinger. He made his Major League debut on August 10, 2011 against the Arizona Diamondbacks. He started the game and allowed four runs in six innings to pick upthe loss. His first win was recorded when he allowed one run in six innings against the San Francisco Giants on August 25. In 10 starts he was 3-5 with a 5.23 ERA.
Sosa signed a minor league contract with the Los Angeles Dodgers in December, 2013 and was assigned to the AAA Albuquerque Isotopes. In seven starts for the Isotopes he was 1-2 with a 3.72 ERA.
Nexen Heroes
On May 15, 2014, he signed with the Nexen Heroes of the KBO.
LG Twins
Sosa signed with the LG Twins of the Korea Baseball Organization for the 2015 season. He re-signed with LG Twins for the 2016, 2017 and 2018 seasons. Since then, he has become one of the most well-known veteran foreign pitchers in the KBO League alongside Dustin Nippert. He became a free agent following the 2018 season.
Fubon Guardians
Sosa signed with the Fubon Guardians of the Chinese Professional Baseball League for the 2019 season.
SK Wyverns
On June 3, 2019, Sosa signed with the SK Wyverns of the KBO League for the remainder of the 2019 season. He became a free agent after the season.
Fubon Guardians (second stint)
On December 13, 2019, Sosa signed with the Fubon Guardians for the 2020 season.
