In the second league match of the 2013–14 season, Rojo scored his first goal of the season, netting Sporting's second goal in their 4–0 win over Académica de Coimbra. On 19 April 2014, Rojo was sent off in a Primeira Liga match against Belenenses, but Sporting held on for a 1–0 win. In an improved season, he contributed six goals as Sporting finished as runners-up.
On 13 August 2014, it was announced that Rojo was being disciplined by his club over his refusal to train after handing in a transfer request to force through a move to Premier League club Manchester United. Sporting president Bruno de Carvalho later alleged that third-party owners of Rojo who were not representatives of Manchester United had pressured his club into selling the player.
On 19 August 2014, an agreement to sell Rojo for €20 million (£16 million) was announced by both Sporting and Manchester United. The next day, United announced that they had signed him on a five-year contract, with Nani moving to Sporting on a season-long loan. Rojo stated that it was a "dream for me" to play with Manchester United.
In June 2014, Rojo was named in Argentina's squad for the 2014 FIFA World Cup. He made his FIFA World Cup debut in their 2–1 victory over Bosnia and Herzegovina at the Maracanã, playing the full match in defence, and contributing to Sead Kolašinac's own goal. In the team's third group match, Rojo scored his first international goal, the winner in a 3–2 victory against Nigeria. The country reached the final before losing to Germany, and Rojo was the only Argentinean named in the Castrol Index Top 11 for the tournament.
In December 2014, Rojo had an affair with a woman he had met at a night club. He accused her of blackmail and took out an injunction to stop newspapers publishing his name in the story. The injunction on his name was lifted in April 2015 due to concerns that other footballers could be suspected of the affair, but it remains in place to prevent the release of images of Rojo.