Cuadrado was signed as an international free agent when he was 17 years old by the Los Angeles Dodgers during the 2014 signing period. He would make his professional debut in 2015 while playing in the Dominican Summer League. The slugger would peak in Double-A Tulsa, in that same year he blasted 10 home runs in 81 games. 

The 6-4 Cuadrado moved on to join the Saitama Seibu Lions in the Japan Eastern League during the summertime and then play in the Venezuelan Winter League with Navegantes Del Magallanes. In 2024 Cuadrado signed with Sultanes de Monterrey, he hit four home runs for the club in 15 games. Most recently Cuadrado was in Venezuela and in 51 games he has a .284 average, 11 doubles, and 10 home runs.  

Between his time in the minor leagues with the Dodgers and playing abroad, Cuadrado has played in 750 games holding a slash line of .254/.332/.393, collecting 119 doubles, 74 home runs, and 386 RBIs.