On 8 November, Cavani scored all four goals, including a half-volley from outside the box and a tremendous free kick, as Napoli came back from 2–1 down to defeat Dnipro Dnipropetrovsk 4–2 in the group stage of the Europa League. Cavani scored a late penalty in the 94th minute to secure a 2–1 victory over Swedish side AIK on 22 November, sending Napoli through to the next round of the Europa League. Cavani was the last player ever to score on the Råsunda Stadium which was the stadium that hosted the 1958 FIFA World Cup final. On 6 January 2013, Cavani netted a perfect hat-trick as Napoli thumped Roma 4–1, gaining ground in the race for the league title as champions Juventus fell to a shock win. Cavani finished the season as leading Serie A goalscorer, with 29 goals, six ahead of Udinese striker Antonio Di Natale in second.
Towards the end of the season, reports emerged that Cavani would leave Napoli, with Chelsea, Manchester City, Paris Saint-Germain and Real Madrid all believed to be interested. On 27 May 2013, however, he told that he was not thinking of moving, saying, "Real Madrid, Chelsea and [Manchester] City are interested in me? I just think about Napoli." Nonetheless, he continued, "If an important offer were to come in, I will talk with president [Aurelio] De Laurentiis." He then paid tribute to his time in Italy, saying, "I'm fine in Italy, I have grown as a man there and as a player with the Serie A experience." Cavani was asked about the interest Chelsea and Manchester City took in him, saying, "I don't know if they made an offer for me, I just know that to be coached by someone like [Man City manager] Manuel Pellegrini or [Chelsea manager] José Mourinho would always be a pleasure." On 23 June, Napoli club president Aurelio De Laurentiis revealed that Cavani's reported £53 million "buyout clause expires" on 10 August 2013.
On 16 July 2013, Cavani joined French champions Paris Saint-Germain on a five-year contract, for a fee believed to be around €64 million, making it the sixth largest transfer in history. The reported sum made Cavani the record signing in French football, breaking Radamel Falcao's €60 million move to Monaco earlier in the summer, and saw him link up with former Napoli teammate Ezequiel Lavezzi in Paris.
At the 2013 FIFA Confederations Cup, Cavani equalised in the semi-final against the hosts and eventual champions Brazil, who eventually won 2–1. In the match for third place, he equalised twice against Italy, his second goal coming from a free kick, taking the game to penalties. Although Cavani scored his spot kick, Uruguay lost; he was named man of the match for his performance.
On 13 November 2013, Cavani scored the last goal in Uruguay's 5–0 away win over Jordan in a play-off for the 2014 World Cup. In their first group game of the finals, against Costa Rica in Fortaleza, Cavani opened the scoring with a penalty after Diego Lugano had been pulled down. Uruguay, however, eventually lost 3–1. Uruguay were eliminated by Colombia, following a 2–0 defeat in the round of 16 on 28 June 2014.
Cavani was married to Maria Soledad Cabris Yarrús, with whom he has two sons, Bautista (born 22 March 2011) and Lucas (born 8 March 2013). In 2013, Cavani announced that he and his wife were to get a divorce, following a split between the two the year before. Then he had a daughter, India (born 17 May 2019), with his girlfriend Jocelyn Burgardt.