On 3 August 2014, Real Madrid triggered Navas' €10 million buyout clause, and he signed a six-year contract with the club. His first competitive call-up was on 12 August, sitting on the bench as Iker Casillas played in the 2014 UEFA Super Cup, which Real Madrid won 2–0 against Sevilla at the Cardiff City Stadium. Navas was given his debut for the club on 23 September, in a 5–1 victory over Elche at the Santiago Bernabéu. He was given his Champions League debut on 26 November, keeping a clean sheet in a 0–1 away win against Swiss side FC Basel. On 20 December, he was on the bench as his team won the Club World Cup, defeating Club Atlético San Lorenzo in the final in Morocco. His performances for Levante in the previous La Liga season and Costa Rica in the 2014 World Cup led him win Player of the Year at the 2014 CONCACAF Awards, earning the dual distinction of the first Costa Rican and goalkeeper to do so.
Navas made his debut in the FIFA World Cup on 14 June 2014, appearing in a 3–1 group stage win against Uruguay in Fortaleza, and started the other two group games only conceding a single goal as Costa Rica qualified for the knockout stages as group winners for the first time.
Navas is a Catholic. He spoke about his faith saying, "God for me comes first. Before every game I kneel, I open my arms and pray ... My favourite passage of the Bible is Galatians 1:10 which says: "If I were still trying to please man, I would not be a servant of Christ." On 3 December 2014, Navas announced that he had obtained Spanish citizenship.