The 19-year-old, who is the son of Sunderland boss Gus Poyet, has signed a four-year deal with West Ham after his contract with the Championship side.
The two clubs have agreed a compensation fee for Poyet due to him being under 24. The midfielder becomes Sam Allardyce's fourth signing of the summer following the arrivals of Aaron Cresswell from Ipswich Town, Cheikhou Kouyate from Anderlecht and Mauro Zarate from Velez Sarsfield.
"I'm very happy to be here. It took me by surprise a little bit and it all happened very quickly," Poyet told the club's official website.
"It's a great club to be at and I know a few of the boys here already.
"It was a no brainer to sign, I wanted to come out here to the training camp as soon as I could and I'm grateful that they've had me here.
"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.
"I played quite a few games last season with Charlton and I just want to push on now. Like any young player, the first thing you want to do is get into the first team. I need to show the staff, players and the gaffer what I'm capable of doing and maybe pushing my way in.
"West Ham is a great club with a lot of tradition, moving to a new stadium and everything, it was a no brainer for me because you never know if you're going to get that kind of chance again."