The 19-year-old, son of Sunderland manager Gus Poyet, was a free agent after his contract at Charlton expired, but the Hammers still needed to agree an undisclosed compensation fee with the Championship club.
Poyet only broke into the Addicks first-team in January and was named player of the year after some impressive displays in their successful fight against relegation.
But the teenager has linked up with his new team-mates at their their pre-season training camp on the French-Swiss border and is keen to prove himself in the Premier League.
"I'm very happy to be here. It took me by surprise a little bit and it all happened very quickly,” he told West Ham’s 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."
Poyet, who has been capped at England U17 level, becomes the club’s fourth summer signing following the arrivals of forward Mauro Zarate, midfielder Cheikhou Kouyate and defender Aaron Cresswell.