The 30-year-old joins the Hammers on a free transfer after his previous deal with Spanish side Real Betis expired earlier this summer.
Rodriguez, who plays at the base of midfield, was part of the Argentina squad which won the World Cup in 2022 and the 2021 and 2024 Copa America titles.
"I feel like it's every footballer's dream to play in the Premier League, and I was very excited when I heard that West Ham were interested in me," Rodriguez said.
Meanwhile, Norwich City have signed Manchester City defender Callum Doyle on a season-long loan.
The 20-year-old travelled with Pep Guardiola's side on their pre-season tour of the United States but is yet to make a competitive first-team appearance for the club.
He previously spent time on loan at Sunderland, Coventry and Leicester, playing 17 times as the Foxes were crowned winners of the second tier Championship last season.
Elsewhere in the Championship, West Brom have signed winger Lewis Dobbin on a season-long loan from Aston Villa.
The 21-year-old moved to Villa in June, having made 15 Premier League appearances for Everton and also spent the 2022-23 campaign on loan at Derby.
West Brom also confirmed the arrival of left-back Gianluca Frabotta from Juventus for an undisclosed fee on a three-year deal.
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