Sosa suffered losses in the early part of his career, the first one came against another future Mexican light flyweight champion Ulises Solís. He faced off against Solis in a rematch in 2003 but lost once again. Sosa would then score fourteen straight victories including a win over former two time minimum weight champion Noel Arambulet.

On April 14, 2007, Sosa won the WBC light flyweight title by besting former champion Brian Viloria by unanimous decision. Sosa has defended his title eight times since then. The most recent one, a 4th-round technical knockout victory over Porsawan Popramook on April 4, 2009.

On November 21, 2009, Sosa lost his title to Rodel Mayol. In the second round, Sosa was badly hurt by a headbutt from Mayol, sending Sosa to the canvas and causing him sustain an injury which was later determined to be a triple-fracture. The referee ruled the headbutt as an unintentional foul and deducted a point from Mayol. After the ringside doctor inspected the wound, the referee allowed the fight to continue despite Sosa still being groggy from the injury. When the action resumed, Sosa was dropped and the referee waved it off moments later. Sosa, who required a metal plate to be implanted on his cheekbone due to multiple fractures, filed a protest with the WBC to have the decision changed to a "no contest". Due to the controversial nature of his defeat the WBC have declared Sosa a "champion emeritus" and he is entitled to either a rematch with Mayol, or a fight with flyweight champion Pongsaklek Wonjongkam, once Sosa has recovered from his injuries. Sosa, who had discussed moving up in weight before his controversial loss, is scheduled to return to the ring on July 10, 2010 in the flyweight division.