Vargas was originally signed by the Minnesota Twins as an amateur free agent in February 2009. He made his professional debut in 2009 with the Gulf Coast Twins. He also played for the Gulf Coast Twins in 2010. Vargas played the 2011 season with the Elizabethton Twins. He was hitting .322/.377/.489 with six home runs over 44 games, when he was suspended 50 games for a violation of the Minor League Drug Prevention and Treatment Program. After his suspension was over in 2012 he played for the Beloit Snappers. In 41 games he had 11 home runs and a 1.030 on-base plus slugging (OPS). Vargas played for the Fort Myers Miracle in 2013. In 125 games he hit 19 home runs. He was added to the Twins 40-man roster on November 20, 2013. Vargas started the 2014 season with the Double-A New Britain Rock Cats. In July he played in the All-Star Futures Game.
Vargas made his MLB debut on August 1, 2014, and hit his first career home run off Jesse Hahn five days later, on August 6, leading the Twins to a 3-1 victory over the Padres.