German connected with Mitchell for a 25-yard touchdown pass in the first quarter and with Wesley Samuels for a 17-yard score in the second as Tennessee State (10-3) built a 17-0 halftime lead.
Tennessee State advances to face conference foe and No. 2-seeded Eastern Illinois on Dec. 7, which beat the Tigers 34-16 earlier this season.
Butler gained just 257 total yards in its first FCS playoff game in program history. The Bulldogs were the first-ever team from the Pioneer Football League to make the playoffs.