Eyestone has signed a three-year contract and will be registered initially with the B team.
The 17 year-old keeper arrives from Duke University, having come through the FC Dallas academy.
Eyestone said: “From my first conversations with Brentford I was really impressed with the recruitment team and the coaches and their approach to working with me to develop my game.
"I can’t wait to get started and I’m laser focused on the work ahead. I’m hungry to contribute to the club’s success and hopefully, one day, play at the highest level."
Brentford B coach Neil MacFarlane also told the club's website: “I am hugely excited about this signing; you can see immediately that he has a great mentality.
“We really look forward to working with him as he has huge potential. The way he speaks and the way in which he carries himself highlights to me the level that he wants to get to.
“He just cannot wait to get started and we cannot wait to get started with him.”