Nyora kupfuurira kweiyi ndima - Kutanga kwake kwepamutemo kwaive nechikwata cheUruguay Wanderers de Durazno. Alzamendi akatamba kuUruguay mu1986 uye 1990 FIFA World Cups, achiwana zvibodzwa vachipesana neWest Germany muna 1986. Akatamba kwemakore akati wandei kuRwizi Plate yeArgentina, achikunda zvese Copa Libertadores de América neIntercontinental Cup
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 ndokuzodzoka kubva kumuchengeti wepagedhi. Gore iroro akatumidzwawo zita rokuti South American Footballer of the Year.