The 19-year-old has signed a four-year deal with the Hammers following his release from Charlton.
The transfer was completed on Tuesday evening after a compensation package was agreed between the two clubs.
"I'm very happy to be here. It took me by surprise a little bit and it all happened very quickly," Poyet told West Ham's official website. "It's a great club to be at and I know a few of the boys here already.
"It's massive to come to a Premier League club. It's what I wanted to do - I wanted a new challenge and soon as I heard there was a chance of stepping up I had to take it."
Poyet leaves Charlton following an eight-year association with the club.