Drexler was born in Montevideo. In 1939 his father, a German Jew, fled to Uruguay with his family at the age of four to escape the Holocaust. His family also fled to Bolivia and lived there. At the time only Bolivia among South American countries was widely open to Jewish immigrants. And over six decades later as his gratitude to Bolivia he made a song, which is included in Bailar en la Cueva. His mother is a Christian of mixed Spanish, French, and Portuguese descent. Drexler was raised Jewish,  but does not follow any organized religions.
Like much of his family, he studied medicine and became an otolaryngologist—an ear, nose and throat specialist. Drexler began playing piano at age five, before attending guitar and composition classes. Although he had an interest in music, he became a doctor like both of his parents. He attended medical school in Montevideo. During his time in medical school, Drexler took a break to hitchhike through Brazil. He also studied music and recorded two albums, which were only released in Uruguay.
In 1995 he was invited to Madrid by well-known Spanish songwriter Joaquín Sabina, who introduced him to other important Spanish singers. Drexler went to Spain to record the album Vaivén in 1996 with Spanish musicians. Vaivén included some old songs from his previous releases mixed with new compositions. He moved to Spain and recorded another four albums: Llueve (1997), Frontera (1999), Sea (2001) and Eco (2004). In 2001, Drexler co-wrote two songs for Spanish singer Rosario Flores ("Agua y Sal" and "Rosa y Miel") for her album Muchas Flores.
Drexler's song "Al Otro Lado del Río" appeared in the internationally acclaimed film The Motorcycle Diaries. Though Drexler himself sang the song on the movie soundtrack, he was not allowed to perform the song at the 2005 Academy Awards, since "he was not popular enough," according to Spanish newspaper El País; Spanish actor Antonio Banderas and Mexican-American musician Carlos Santana sang the track instead. Upon winning, Drexler recited two verses of the song at the podium. Drexler became the first Uruguayan to win an Academy Award, and this was also the first for the country.
After that, he released 12 Segundos de Oscuridad (2006); this album contained ten original songs and two covers: "High and Dry" from British band Radiohead and "Disneylandia" from Brazilian Titãs. Although he lives most of the year in Spain, his albums were partially recorded in Uruguay with Uruguayan musicians. Juan Campodónico and Carlos Casacuberta, former members of rock band El Peyote Asesino, had produced Drexler's albums from Frontera to 12 Segundos De Oscuridad. In 2008, he released a double live album, recorded in diverse concerts in Spain: Cara B (2008), mainly filled with songs previously unrealeased. During 2009, Drexler worked with Colombian performer Shakira on the Spanish-language versions of her singles "She Wolf", "Did it Again" and "Waka Waka (This Time for Africa)", and recorded Amar la Trama (2010) in Madrid, Spain in just four days, with musicians playing live on studio. Drexler described the album as playful, without "the melancholy and anguish" of 12 Segundos.
His music is a combination of Uruguayan traditional music (candombe, murga, milonga, tango), bossa nova, pop, jazz and electronic music, which results in very personal compositions with original arrangements. The words also play an important role in his songs. Apart from love, reflections about identity, race and religions are a constant in his work.
^"Ecos de Drexler" (in Spanish). Montevideo COMM. Desde hace mucho tiempo he perdido mi fé en todas las religiones organizadas. No creo en Dios.Al menos no de la manera en que se lo describe habitualmente.