DeBesse attended Stratford High School in Houston, Texas and earned his bachelor of science in Education at the Texas State University-San Marcos in 1982. He began his coaching career as a graduate assistant at Texas State, before being hired as an offensive assistant to Jim Wacker at Texas Christian University. In 1990 he was appointed offensive coordinator. DeBesse followed Wacker to Minnesota in 1992. He succeeded Jim Bob Helduser as head coach at his alma mater in 1997. After mediocre success, DeBesse was replace by Manny Matsakis in 2003. He joined Joe Tiller's staff at Purdue as wide receivers coach.