Puig played for the Cuban national baseball team in the 2008 World Junior Baseball Championship, winning a bronze medal. He then played for the Cienfuegos team of the Cuban National Series in the 2008–09 Cuban National Series. He batted .276 with five home runs in his debut season. Puig enjoyed a breakout season in the 2009-10 Cuban National Series, with a .330 batting average, 17 home runs, 47 runs batted in (RBIs) and 78 runs scored in 327 at-bats. Puig also played for the Cuban national team in the 2011 World Port Tournament, where he tried to defect along with teammate Gerardo Concepcion. Concepcion was successful while Puig was not, and he was disciplined by not being allowed to play during the 2011–12 seasons.
Since 2011, he tried to defect to Mexico five times, in order to become a legal resident so he could become eligible to enter the MLB or ULB Draft. He defected from Cuba in 2012, and was entered into the ULB Draft after being awarded a visa.
Puig was selected 1st Overall in the 2013 ULB Draft by the Green Bay Blizzard. Puig is likely to be assigned to the American Association’s Rochester Red Wings, the Triple A affiliate of the Blizzard to start the 2013 season. Many scouts suggest Puig could make an immediate impact, and will likely debut in 2013.