Navas was part of the Costa Rica national team that played in the 2003 FIFA U-17 World Championship held in Finland. He was called up for the senior national team for the first time in August 2006, to play a friendly tournament in Europe against Austria and Switzerland. He was a member of the team that won a bronze medal at the 2006 Central American and Caribbean Games. Navas gained his debut senior cap on 11 October 2008, in a 4–1 away victory over Suriname in the third round of qualification for the 2010 FIFA World Cup, which qualified the team into the next stage. Four days later at the Estadio Ricardo Saprissa, he kept his first international clean sheet in a 2–0 victory over Haiti. The team eventually reached the intercontinental play-offs for the tournament, losing narrowly to Uruguay. Navas was also a member of the Costa Rica squad that finished runners-up in the 2009 UNCAF Nations Cup.

Born in San Isidro de El General, Navas made his professional debut with Deportivo Saprissa on 6 November 2005, in a Primera league match against Asociación Deportiva Carmelita. He was the first-choice keeper in his two final seasons with the club, winning six national championships and the 2005 CONCACAF Champions’ Cup. Navas was also part of the Saprissa team that finished in 3rd place at the 2005 FIFA Club World Championship.

Navas appeared with Los Ticos in two CONCACAF Gold Cup competitions, being named the best goalkeeper in the 2009 edition as he helped his country reach the semi-finals. He returned to help them to the quarter-finals two years later, but missed the 2013 tournament through injury, suffering the same fate again in 2015.

In July 2010, Navas signed with Albacete Balompié of the Segunda División in Spain exactly 20 years after countryman Luis Conejo, who occupied the same position. He played in 36 games out of 42 during his first season, but his team suffered relegation after finishing in last position.

In the 2011–12 campaign, Navas reached La Liga, being loaned to Levante UD for a year. He made his debut in the competition on 13 May 2012 in the last round, playing in a 3–0 home win against Athletic Bilbao that confirmed the club’s first ever qualification for the UEFA Europa League, and joined the Valencian club on a three-year contract in July of that year. Despite initially being back-up to Gustavo Munúa, Navas was the first-choice goalkeeper for Levante in the 2013–14 season, and was nominated as the league’s best goalkeeper alongside Thibaut Courtois of Atlético Madrid and Willy Caballero of Málaga CF, going on to win the prize. In March 2014, he was La Liga Player of the Month, the first goalkeeper to win the accolade. He finished the season as the goalkeeper with the most saves (267) in La Liga, finishing fourth in the Zamora Trophy conceding 39 goals in 36 games despite playing for a ‘lower ranked’ team.

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.

With the departure of Casillas from Real Madrid, Navas inherited his number 1 shirt for the 2015–16 La Liga season. Madrid agreed to transfer Navas to Manchester United in part-exchange for David de Gea on 31 August 2015, but the deal collapsed because the documents were not submitted to FIFA before the Spanish transfer deadline.

He was the starting goalkeeper when Madrid won the 2016–17 La Liga and the 2016–17 UEFA Champions League. On 29 January 2017 after the 3–0 win at home to Real Sociedad in La Liga, Navas became the first player from Costa Rica to reach 100 La Liga appearances. His performances during the double trophy winning season led him to win the Player of the Year for the second time at the 2017 CONCACAF Awards.

Often rated by several pundits as one of the best goalkeepers in the world, and considered by some in the sport as the best in the history of CONCACAF, Navas is an energetic keeper, who is primarily known for his agility, speed, and athleticism in goal, as well as his quick reflexes and excellent shot-stopping ability, which enables him to compensate for his relative lack of height. He has also drawn praise in the media for his decisive performances in important matches, and his penchant for producing difficult and crucial saves for his team in key moments. In 2017, former goalkeeper Manuel Almunia praised Navas, describing him as “the prototype model” goalkeeper, also adding: “He’s spectacular between the posts, he’s brave, skillful and very agile. He maintains his concentration and he’s also intelligent… For me he deserves all the respect he gets for what he’s done.”

A documentary film, Hombre de Fe (Man of Faith), based on the life of Navas was released on 28 December 2017.

Navas’ fifth season at Real Madrid began with the much-anticipated arrival of Thibaut Courtois to compete for the number 1 goalkeeping spot. Despite Courtois’ arrival, Navas was selected to play in the 2018 UEFA Super Cup which resulted in a 2–4 loss to city rivals Atlético Madrid. Navas then went on to start the opening game of La Liga, comfortably keeping his first clean sheet of the season with a 2–0 win over Getafe CF. His performances in the last Champions League campaign led him to be voted as the 2017–18 UEFA Champions League Goalkeeper of the Season beating off competition from then Roma goalkeeper Alisson Becker and Juventus goalkeeper Gianluigi Buffon. On 9 January 2019 Navas made his 150th appearance in all competitions for Madrid in their 3–0 home win against Leganés in the Copa del Rey. On 6 April 2019 Navas became the first ever non-Spanish goalkeeper to reach 100 La Liga appearances for Real Madrid.

In May 2018, Navas was named in the final squad for the 2018 World Cup. He started all three group games making 10 saves and conceding five goals.

Navas joined Paris Saint-Germain on a four-year contract on 2 September 2019, becoming the inaugural Costa Rican to play for the men’s team. He made his Ligue 1 debut 12 days later, keeping a clean sheet in a 1–0 home win against RC Strasbourg. On 24 July 2020, Navas won his 20th career club trophy after beating Saint-Étienne in the Coupe de France final. On 31 July, he kept a clean sheet and saved a spot kick in the resulting shoot-out as PSG defeated Lyon 6–5 on penalties in the 2020 Coupe de la Ligue Final. That win marked Navas’ 15th trophy in Europe, matching Rafael Márquez’s record for a CONCACAF player. On 23 August 2020 Navas started his 4th Champions League final losing to Bayern Munich 1–0. Navas finished the European campaign with 5 clean sheets, the second most after Manuel Neuer despite having played 191 minutes less.

Navas started his second season with the Parisian club as one of the 7 club players to test positive for COVID-19. He missed the clubs first two games due to the quarantine requirement. On 16 September 2020 Navas played his first game of the season winning and keeping a clean sheet against Metz. On 17 September 2020 UEFA announced Navas, for the second time, as a nominee for the 2019/20 Champions League Goalkeeper of the Season award. On 20 September Navas starred in PSG’s 0–3 away win to Nice with a second conescutive clean sheet as well as several excellent saves.

In 2020, Navas cited compatriot Lester Morgan as his goalkeeping idol.