The 46-year-old replaces Oscar Tabarez, who was sacked last month after 15 years in the job.
Tabarez's departure came after a string of poor results that left the two-time World Cup winners seventh in South America's qualifying group for Qatar 2022.
The top four qualify automatically for the finals and the fifth-placed side go into an inter-continental playoff.
His longest coaching spell was at Mexican side Pachuca, who he led to the Mexican championship in 2016. He also won the CONCACAF Champions league with both Pachuca and Monterrey.
The Uruguayan Football Association did not provide details about his contract but local media said Alonso had signed a deal at least until the end of the South American qualifiers.