Brown is getting a six-year, $19.05 million contract, according to the school.
His contract will make him the highest-paid new Big 12 coach, ahead of three other league schools that made hires recently: Kansas (Les Miles), Kansas State (Chris Klieman) and Texas Tech (Matt Wells).
"When I started this national search, I learned very quickly that he checked all the boxes of what I was looking for in our next football coach," athletic director Shane Lyons said in a statement. "There is no question that the Mountaineers are West Virginia's team, and I know our fans are going to love his energy, passion, work ethic and excitement that he is going to bring to our program."
Brown, 38, is one of only six FBS coaches with 10 victories in each of the past three seasons.
A disciple of Hal Mumme's and Mike Leach's Air Raid offenses, Brown has a 35-16 record in four seasons at Troy. After going 4-8 in his first season, the Trojans are 31-8 the past three seasons combined and won three bowl games. They won the 2017 Sun Belt Conference title and were co-Sun Belt East champions this season.
In 2017, the Trojans upset No. 25 LSU 24-21 to end the Tigers' 49-game nonconference winning streak at Tiger Stadium.
The move comes only two days after former Mountaineers coach Dana Holgorsen was officially hired at Houston. Holgorsen went 61-41 in eight seasons at West Virginia.
ESPN's Jake Trotter contributed to this report.