Narváez grew up in El Playón de San Francisco, a village in Ecuador's Sucumbíos province close to 3,000 metres above sea level. He started racing with a school cycling club run by one of his teachers, Juan Carlos Rosero, a former Olympic rider who also brought through Richard Carapaz and Jonathan Caicedo. He won Pan American junior titles in 2015 and the Circuit des Ardennes in 2017.
He signed a three-year deal with Quick-Step Floors, rode one season on it, then broke the contract in late 2018 to move to Team Sky for 2019, the season the squad was rebranded as Ineos. His first win at that level came at the 2020 Giro d'Italia, on the loop out of Cesenatico, Marco Pantani's home town, where he went clear from the day's breakaway and finished more than a minute ahead of Mark Padun.
Through the early 2020s he raced the cobbled classics as well as the Giro, taking sixth at both Strade Bianche and E3 in 2022. In 2023 he won the Tour of Austria overall with three of the five stages, then took the Pan American Games road race in Chile in a sprint from a small group. The Ecuadorian national road title came that year too, and again in 2024 and 2025.
The 2024 Giro started at Venaria Reale and finished its opening day in Turin. Narváez won it from a reduced group ahead of Tadej Pogačar and wore the maglia rosa for a day. He had been second at the Tour Down Under that January, nine seconds behind Stephen Williams. UAE Team Emirates signed him for 2025 on a two-year contract.
He went back to Adelaide in 2025 and won the Tour Down Under by nine seconds, the margin built on a stage win up Willunga Hill. Much of the season went on work for Pogačar, including a spell on the front over the Cipressa at Milan-San Remo. He still won a stage of the Deutschland Tour, finished second overall there, and came thirteenth at the Tour de France.









