Weah, 17, has signed a four-year contract that includes a club option for a fifth year.
"It's another chapter in us showing that we are committed to the youth development," United manager Adrian Heath said in a club statement. "As I said last year when we signed Fred [Emmings], if there was more kids that we felt were ready to make the step, we would have done it. Patrick [Weah] is another one of them kids.
"He's got loads of raw, natural talent. We have to harness that over the next few years, and then try and teach him the sort of nuances of the game, because of all the physicality in the game. And the basics, he's got in abundance."
"We're delighted he's decided to join us," Heath said. "It's the start of a lot of hard work for him but we feel as though he's got a lot of raw attributes to make the next step. Can we continue this now, can we continually bring one or two players from the system below, to keep promoting them to the first team squad? That's got to be the aim."