Alonso was born in Montevideo, the country’s capital. He made his professional debut with C.A. Bella Vista where he remained four seasons, helping the side to the second division title in 1997.

Alonso made eight appearances for the Uruguay national team, during five years. His debut came on 17 June 1999 in a 3–2 friendly win over Paraguay, in Ciudad del Este.

In 2006, the 31-year-old Alonso returned to his country and joined hometown’s Club Nacional de Football. However, shortly after, he moved abroad again, signing with Shanghai Shenhua F.C. in the Chinese Super League; after two slow seasons with Gimnasia, the veteran switched to Peñarol.

Alonso started working as a manager in 2011, being in charge in quick succession of Bella Vista, Club Guaraní, Peñarol and Club Olimpia (the second and fourth sides from the Paraguayan Primera División). In December 2014, he was appointed at C.F. Pachuca of the Mexican Liga MX, leading them to the 2016 Clausura and the subsequent edition of the CONCACAF Champions League. On 4 May 2018, he was released.

On 2 June 2018, Alonso was named coach of C.F. Monterrey. In May of the following year, after the 2–1 aggregate victory over Tigres UANL in the Champions League final, he became the first manager to win the competition with two different clubs. On 30 September 2019, following a 0–2 loss to the same opposition that left his team in 12th place on the general table, he was dismissed.

On 30 December 2019, Alonso was announced as the inaugural head coach of Major League Soccer side Inter Miami CF.