The Argentine's move from Deportivo Maldonado was confirmed on Tuesday afternoon.
The 22-year-old returned from Olympic duty with Argentina to complete a season-long switch to the London Stadium, and could go straight into the squad to face Astra Giurgiu in the Europa League play-off first leg on Thursday.
Calleri's move has been a long time in the making, with international clearance and his involvement in Rio de Janeiro delaying the transfer.
However, he is now officially a West Ham player, with the Hammers reportedly having paid £4 million to sign him for a single season, and the 2015 Copa Libertadores leading scorer admitted to the club's website that he is delighted to be playing in the Premier League.
He said: "I am very happy to have signed for West Ham. This represents a fantastic new challenge for me and I am really looking forward to it. The Premier League is the best league in the world and so I am delighted to have this opportunity to play in such a prestigious league.
"It is a new chapter for me and so I am excited to get started here and to fit in as quickly as possible, so that I can be as useful as possible to my new teammates and manager.
"These are exciting times for me personally and I am motivated to adapt my game to this new environment. I am a player who always tries to work as hard as I can and to do my best for the team."
Calleri will add to Slaven Bilic's attacking options of Andy Carroll, Diafra Sakho, Enner Valencia, and summer signings Andre Ayew and Ashley Fletcher.
He will be hoping to continue his recent impressive goal record in South America and make an immediate impact in East London.