The teenage goalkeeper has signed for a £12m million fee from Major League Soccer side Chicago Fire.
Slonina has signed a six-year contract with the club.
"Honoured to join such a massive club with so much history," he tweeted.
"It's a privilege to wear the badge of the best side in London. Thank you to all the fans for the warm welcome. Can't wait to get started.
He added: "Thank you to everyone at the club for making a childhood dream come true & thank you to the fans for an amazing warm welcome. The hard work starts now."