Pedzisa mutsara unotevera: Kutanga kwake chaiko kwakanga kuri nechikwata cheUruguay cheWanderers de Durazno. Alzamendi akatamba kuUruguay mu1986 uye 1990 FIFA World Cups, achiwana zvibodzwa vachipesana neWest Germany muna 1986. Akatamba makore anoverengeka kuRwizi Plate yeArgentina, achikunda zvose zviri zviviri Copa Libertadores de América uye
Mukombe weIntercontinental muna 1986. Mumutambo wekupedzisira weIntercontinental Cup, Alzamendi akatumidzwa zita remurume wemutambo uye akawana chinangwa chega chemutambo muminiti yechi28 paakatungamira mumambure mushure mekunge pfuti yake yekutanga yarova mbiru uye ndokuzodzoka kubva kumuchengeti wepagedhi. Gore iroro akatumidzwawo zita rokuti South American Footballer of the Year.