Cuadrado made his senior international debut in 2010, and has since earned over 35 caps. He played at the 2011 Copa América, and was also part of the Colombian squad which reached the quarter-finals of the 2014 FIFA World Cup.

Early career
Born in Necoclí, Cuadrado began his club career in Atlético Uraba, where his preferred position was as a forward. However, on the recommendation of club founder Nelson Gallego, Cuadrado took on a midfielder role, usually as a winger. Cuadrado's confidence and ability in the position also allowed him to play occasionally as a full-back. He later had a brief spell at Colombian second division club Rionegro, where he languished on the bench under coach Víctor González Scott, but his potential was recognized by Juan José Peláez of Independiente Medellin. He was eventually signed by Medellin in 2008, allowing him to return to the first division. That year, Cuadrado made his senior professional debut against Boyaca Chicó, a match in which he would also score his first goal. Cuadrado lasted until 2009 at Medellin before making the leap to European football.
Udinese
On 2 July 2009, Cuadrado signed for Italian Serie A side Udinese on a five-year deal for an undisclosed fee. He played his first match for Udinese in a 2–1 win with Chievo Verona as a right wing back. However, Cuadrado was not given many first-team minutes the rest of the season, and he would be sent on loan to fellow Serie A side Lecce.
Loan to Lecce
On 3 August 2011, it was officially confirmed that Cuadrado would join Lecce for the 2011–12 season. He scored his first goal for the club against Cesena, the sole goal in a 1–0 victory. In spite of further impressive performances from Cuadrado, however, Lecce finished 18th, and was relegated from Serie A. He would then return to Udinese to avoid playing in the second flight.
Fiorentina
On 23 July 2012, Cuadrado completed a move to Fiorentina. He scored his first goal for the Florence club in a 4–1 victory at home to Cagliari. He finished the season with 5 goals and 6 assists, an integral part of a side that finished 4th in Serie A.
On 11 February 2014, Cuadrado scored Fiorentina's second goal of a 2–0 Coppa Italia semi-final second leg victory over his former club, Udinese, securing La Viola a 3-2 aggregate victory and a spot in the final. He was suspended for the final, which his team lost 1–3 to Napoli on 3 May.
On 16 June 2014, amid media speculation of a move for the player to a larger club, Fiorentina agreed a deal to purchase full ownership of Cuadrado's contract from Udinese.
International career
After being transferred to Udinese, Cuadrado received his first call-up to the Colombia national football team, as well as a spot in the starting XI in a match against Venezuela on 3 September 2010. He was named man of the match and scored the opening goal of a 2-0 victory.
Cuadrado was called up by José Pekerman for a friendly against Mexico on 29 February 2012 at Miami Gardens. He started the match and scored the second goal, a tap-in from a Pablo Armero pass, for Colombia in a comfortable 2-0 victory.
Cuadrado was named in Colombia's 23-man squad for the 2014 FIFA World Cup. In the third group stage match, a 4–1 defeat of Japan, he scored the team's opening goal from a penalty kick. He would end his World Cup campaign with 4 assists, tying with Toni Kroos as the competition's assist leader.