He studied to become an otolaryngologist (i.e., ear, throat, and nose doctor) like his parents but ultimately chose to pursue his passion for music. Drexler made his full-length album debut with La Luz Que Sabe Robar (1992), followed by Radar (1994). In hopes of furthering his career, he went to Spain in 1995 upon the invitation of legendary singer/songwriter Joaquín Sabina, who showed him around the Madrid music scene and helped him secure an international recording contract.

So impressed with Madrid that he chose to make his permanent residency there, Drexler got together with a band of Spanish musicians to re-record material from his first two albums along with new material and release it internationally as Vaivén (1996), the first of four albums on Virgin España. The highlights of Vaivén along with those of its three follow-ups, Llueve (1998), Frontera (1999), and Sea (2001), were later compiled on the best-of compilation La Edad del Cielo (2004).

Upon his switch to the Dro Atlántic label, Drexler's career reached new heights of success with the critically acclaimed Eco (2004). Most notably, the album includes the Oscar-winning song "Al Otro Lado del Río," which was prominently featured during the closing credits of the Walter Salles film The Motorcycle Diaries (2004). Eco proved so successful, in fact, that it was re-released with bonus material as Eco² (2005). Subsequent albums include 12 Segundos de Oscuridad (2006), the double-disc live album Cara B (2008), and Amar la Trama (2010).

Drexler spent most of 2011 and 2012 touring. In 2013 he completed writing songs for a new album and decided to record in Colombia, with co-producers Carlos Campón and Sebastián Merlín. The resulting collection of 11 new songs was entitled Bailar en la Cueva and issued in March of 2014. The album featured guest appearances by Caetano Veloso, Ana Tijoux, Bomba Estéreo, and Eduardo Cabra of Calle 13. ~ Jason Birchmeier