Brown resigned last month after guiding the Longhorns for 16 seasons.
Texas Athletic Director Steve Patterson said Strong would "build on the proud tradition of Texas football and the 16 great years that Mack Brown gave to the program."
The AD praised Strong's efforts at Louisville, where he led "a resurgence ... with double-digit game winning seasons, and twice been selected conference coach of the year."
"I'm excited and my family is excited to have the chance to lead one of the premier football programs in the country," Strong said in a statement. "Texas is one of those places that is always on your radar and a program anyone would dream of being a part of because you have a chance to compete on a national level every year."
Louisville compiled an impressive 23-3 under Strong over the past two years, capturing two Big East Conference championships and posting a 3-1 record in bowl games during his tenure.