Navas was part of the Costa Rican squad that played in the 2003 FIFA U-17 World Championship held in Finland. He was called up for the full 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 first 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. 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. 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 very first time. On 29 June, in the round of 16 match against Greece, Navas was named man of the match after several saves in normal time and a save from Theofanis Gekas’ kick during the penalty shootout, which led to Costa Rica reaching the quarter-finals for the first time ever. He ended the tournament with three clean sheets from five matches, as the country was knocked out by the Netherlands after a penalty shootout, where he was named man of the match for the third time, the only goalkeeper to do so. He was also one of three nominees for the Golden Glove Award, losing out to Manuel Neuer of Germany. In May 2018, Navas was named in the final squad for the 2018 FIFA World Cup. He started all three group games making 10 saves and conceding 5 goals.