In 2006, Asenjo was selected to the Spain under-17 side as they took home the bronze medal in the UEFA European Championship. He started in all of the matches, except the third-place play-off.

Born in Palencia, Castile and León, Asenjo started his playing career in the academy of Real Valladolid and soon emerged as one of the most talented players in the club. Succeeding first-team goalkeepers Alberto and Ludovic Butelle, he was promoted to the senior side during the 2007–08 season, and kept a clean sheet in a 2–0 home win over Villarreal on 2 December 2007 which marked his La Liga debut; he conceded just once in his first five matches, which earned him a permanent starting berth and saw him rewarded with his first professional contract in January of the following year.

On 8 July 2009, Asenjo transferred to Atlético Madrid for a reported fee of €5 million and signed a four-year deal with the club. He started the season as first-choice, but often alternated with compatriot David de Gea.

In May 2010, Asenjo suffered a ruptured anterior cruciate ligament and was ruled out for a further six months. Upon his return, he found his place in the squad uncertain and, in December of that year, he was loaned to Málaga until the end of the campaign, with teammate Ignacio Camacho also making the move in a permanent deal.

On 6 February 2011, during a league match between Málaga and Sevilla, Asenjo suffered a second ACL injury to his right knee when he turned awkwardly on the pitch surface. He missed the remainder of the season, having only made five league appearances during his loan spell.

With his game time limited in Madrid, Asenjo agreed to a one-year loan deal with Villarreal in July 2013 with the club retaining a buyout option at the end of the season. He excelled during his first year, missing only three league matches and keeping nine clean sheets as the side qualified for the play-off round of the UEFA Europa League; the ensuing summer, he signed a permanent five-year contract.

Asenjo retained his starting spot throughout the 2014–15 campaign but, on 29 April 2015, in the dying minutes of a 1–0 loss to former side Atlético, he tore his ACL for a third time while attempting to make a save. After an excruciating rehabilitation process, he made his return to the squad on 23 February 2016 when he was named on the bench by Marcelino García Toral for the Europa League fixture against Italy’s Napoli. The following month, and in the same competition, he played his first game in nearly 11 months and kept a clean sheet in a 0–0 draw (2–0 aggregate win) at Bayer Leverkusen.

In March 2015, Asenjo received his first ever callup to the senior national team, being named in Vicente del Bosque’s squad for a UEFA Euro 2016 qualifier against Ukraine and a friendly with the Netherlands. He did not make his debut until 29 May the following year, in a 3–1 friendly win over Bosnia and Herzegovina at the AFG Arena in St. Gallen, Switzerland; he was the first footballer from Palencia to win a cap since Jesús Landáburu 36 years prior.

Asenjo reclaimed his starting position following his recovery but on 26 February 2017, during the first half of a league home match against Real Madrid, he suffered a ruptured anterior cruciate ligament for a fourth time, this time in his left knee– prior to the misfortune, he had conceded the fewest goals in the league and had kept 11 clean sheets. He spent nine months on the sidelines as a result of the injury, before making his return in November in a Copa del Rey encounter against Ponferradina.