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 or wing-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 Medellín. 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 Boyacá Chicó, a match in which he would also score his first goal. Cuadrado remained in Medellin until 2009, before making the leap to European football.

On 2 July 2009, Cuadrado signed with Serie A club Udinese on a five-year deal for an undisclosed fee. He played his first match for Udinese in a 2–1 win against Chievo as a right wing back. Cuadrado, however, was not given many first-team minutes for the rest of the season, and he would be sent on loan to fellow Serie A side Lecce.

After being transferred to Udinese, Cuadrado received his first call-up to the Colombia national team, as well as a spot in the starting XI in a match against Venezuela on 3 September 2010, scoring the opening goal of a 2–0 victory at the Estadio José Antonio Anzoátegui in Puerto La Cruz.

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 to Serie B. He would then return to Udinese. Around this time, he was watched by an unidentified English club who deemed that at 23 years he was too old to be signed by the club.

On 23 July 2012, Cuadrado completed his move to Fiorentina, initially in a temporary deal for a €1 million transfer fee. He scored his first goal for the Florence club in a 4–1 victory at home against Cagliari. He finished the season with five goals and six assists, an integral part of a side that finished fourth in Serie A. In June 2013, La Viola purchased half of the registration rights of Cuadrado for €5 million.

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 booked after scoring the winning goal, and 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 to a larger club by the player, Fiorentina agreed a deal to purchase full ownership of Cuadrado’s contract from Udinese, for an additional €12 million.

On 2 February 2015, Premier League club Chelsea announced the signing of Cuadrado from Fiorentina on a four-and-a-half-year deal, for a reported initial fee of £23.3 million, which could potentially rise to £26.8 million; Mohamed Salah went the other way on loan. While in Fiorentina’s financial filing of year 2014, in the management report (Italian: Relazione sulla Gestione), Fiorentina declared the fee was €30 million. Upon signing, Cuadrado said, “I am very happy and thankful for this opportunity I’ve been given. This is a great club and honestly it is like a dream to join the Chelsea family and to know that the manager (José Mourinho) believes in me. I’m happy.” His Fiorentina manager, Vincenzo Montella, stated, “It was a sale that had to be done. Cuadrado is a player that has always arrived with a smile and given everything to Fiorentina – I’m happy for him because he is going to a big club.” Mourinho commented that he was not in a “rush” for Cuadrado to fully showcase his abilities within a short time, as stated in an interview: “He needs time to adapt to the Premier League. I know Italy and I know the difference between Italy and England; his formation, his development, his experience – everything was in Italy.”

On 25 August 2015, Cuadrado signed a season-long loan deal with Juventus for €1.5 million. He was issued squad number 16. Five days later, he made his debut, as a 75th-minute substitute in a 2–1 defeat away against Roma. On 31 October, Cuadrado scored his first Juventus goal against Torino to secure a 2–1 win in extra time at the end of the Derby della Mole. On 16 March 2016, Cuadrado scored his first ever UEFA Champions League goal in a 4–2 away defeat against Bayern Munich, in the second leg of the round of 16. On 21 May, he came off the bench to set-up Álvaro Morata’s match-winning goal against Milan in the 110th minute of extra time of the 2016 Coppa Italia final, in Rome’s Stadio Olimpico; Juventus’ 1–0 victory enabled the club to secure the domestic double for the second consecutive season.

On 13 October 2015, Cuadrado was sent off at the end of Colombia’s 3–0 loss away against Uruguay in 2018 FIFA World Cup qualification, for elbowing Diego Rolán.

Cuadrado returned to Chelsea on 30 June 2016 and played a few friendlies for Chelsea, including the 2016 International Champions Cup. On 31 August, however, Cuadrado returned to Juventus on a three-year loan for €5 million per season, plus a conditional obligation to buy the full registration rights of Cuadrado from Chelsea for an additional €20 million (or €15 million and €10 million in the second and the third year of loan respectively), after certain sports related achievements. Chelsea would also receive a maximum of €4 million bonuses, making the fee potentially raise to €29 million. He was assigned the number 7 shirt at this time. Cuadrado scored his first league goal in nine months on 5 February 2017, scoring the only goal in a 1–0 home win in a Derby d’Italia clash with Inter Milan.

On 22 May 2017, it was announced that Cuadrado had joined Juventus on a permanent deal for a preset price of €20 million until 2020, after a clause in his contract had been triggered (winning the Serie A mathematically in round 37). In the 2017 UEFA Champions League Final against defending champions Real Madrid on 3 June, Cuadrado came on as a substitute for Andrea Barzagli in the 60th minute, but was later controversially sent off after picking up two yellow cards, the second for an alleged stamp on Sergio Ramos; Juventus lost the match 4–1.

Cuadrado is known for his direct, offensive style of play along the right flank, and is capable of both scoring goals himself as well as creating goalscoring opportunities for his teammates. He is capable of playing both as a wide midfielder, or as a winger in an attacking trident, due to his ability to make attacking runs down the wing. He often utilises his speed, flair, and technical skills to beat opponents in one on one situations and create space and time for him to shoot on goal, or deliver crosses to strikers in the area. Due to his playing style, which frequently involves taking on full-backs on the flank, and subsequently running towards the touch-line to deliver balls into the area, he has been labelled one of the few “traditional” wide midfielders in the current game. Tim Vickery, a reporter on South American football, stated that as Colombia’s defensive players played near the goal in order to cover for veteran centre-back Mario Yepes, Cuadrado’s pace, agility, creativity and dribbling ability on the right wing moved the team forward at the 2014 World Cup. Vickery also added that, in addition to his main role on the wing, Cuadrado is also adept as an attacking right back or wing back, due to his energy and work rate, as well as his other aforementioned qualities; he has also been used in this position more recently at Juventus, under managers Massimiliano Allegri, and Maurizio Sarri. A versatile player, he is also capable of playing on the left, a position which allows him to cut into the middle and shoot on goal with his stronger, right foot. In his early career, he was initially deployed as central or defensive midfielder, and was even deployed as an outright forward or striker; he is also capable of playing as an attacking midfielder, or as a second striker on occasion, and has even been used as a more offensive-minded central midfielder at times, known as the mezz’ala role, in Italian (which literally translates to “half-winger”). Despite his talent and ability, however, Cuadrado’s decision-making and reading of the game have been criticised at times in the media; moreover, he has also drawn criticism at times from some in the sport for being inconsistent, and for having an overly individualistic playing style. In 2017, Mina Rzouki of ESPN FC noted that: “Cuadrado has yet to learn the art of making decisive runs on a consistent basis. Still naive and capable of slowing down a game despite his pace, he does not boast the football intelligence of a world-beater.”

Following the arrival of Cristiano Ronaldo from Real Madrid in 2018, Cuadrado allowed Ronaldo to wear the number 7 shirt, with Cuadrado reverting to number 16, saying on Instagram “It is better to give than to receive […] Blessings to Cristiano in this new adventure”.

On 21 September 2019, Cuadrado made his 100th Serie A appearance with Juventus in a 2–1 home win over Verona, during which he won a penalty which was converted by Cristiano Ronaldo. During the 2019–20 season, under the club’s new manager Maurizio Sarri, Cuadrado was often deployed as an attacking right-back. On 3 November, he made his 150th appearance for the club in a 1–0 away win over cross-city rivals Torino in the Derby della Mole. Later that month, he extended his contract with the club until 2022.

On 30 May 2019, Cuadrado was included in the 23-man final Colombia squad for the 2019 Copa América.