The 22-year-old midfielder, who came through the youth system at famed Argentine club River Plate, has been capped four times at under-20 level.
“This is very emotional because it is one of the best teams I have ever signed for," he said on the club website.
"It is a very important step for my career and for my future."
Lanzini becomes West Ham’s seventh close-season signing following the arrivals of Dimitri Payet, Angelo Ogbonna, Pedro Obiang, Carl Jenkinson, Stephen Hendrie and Darren Randolph.
(Reporting by Toby Davis; editing by Ken Ferris)
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