Goal revealed in early January that a deal was on the cards for the 18-year-old centre-half.
Confirmation has now arrived, with the youngster penning a four-and-a-half-year deal with the Bundesliga champions.
Richards was initially loaned to Bayern’s U19 side in the summer of 2018 and has impressed during his time in Germany.
He formed part of the club’s senior squad at the 2018 Audi Summer Tour and has now been tied to a contract which is due to run until 2023.
“We are very happy that we have agreed Chris’ transfer from FC Dallas," Bayern’s campus manager, Jochen Sauer, told the club’s official website.
“Chris has represented us very well in the last six months and we are convinced that he will continue to develop well here at FC Bayern. We see great potential in him.”
U.S. U20 international Richards is looking forward to opening another exciting chapter in his promising career, with big things expected of him as his undoubted potential continues to be unlocked.
He said of agreeing a permanent stay at the Allianz Arena: “I've always dreamt of playing for a big European club and now I’m at one of the world’s top teams. I'm really happy.”
Bayern have had an elite player development program partnership in place with FC Dallas since February 2018.
As they continue to benefit from that agreement, the club’s preside of the Americas, Rudolf Vidal, said: “Just under a year ago, we entered a partnership with FC Dallas to create pathways to professional soccer in the U.S. and Europe for young players.
“To be able to announce the transfer of a player today is a landmark moment for all involved.”
Richards becomes the second player developed in MLS to have linked up with Bayern this season.
Canada international Alphonos Davies is also part of the fold in Bavaria, with the 18-year-old winger now forging a career in Europe after making a record-setting $13.5m (£10m) move from the Vancouver Whitecaps.