The run by the Racers (3-6, 2-4 Ohio Valley Conference), who ended a two-game losing streak, was capped by a 21-yard interception return for a touchdown by Austen Lane in the fourth quarter.
Lane also had two sacks to become Murray State's all-time leader.
Brockman completed 18 of 33 passes for 216 yards, despite being sacked five times.
Terrence Holt scored two touchdowns for the Governors (3-6, 2-4), including a 99-yard kickoff return and a 30-yard run.
Derrick Townsell caught five passes for 55 yards to become Murray State's career receptions leader with 147. Charlie Jordan rushed for 69 yards to become the 10th Racer to reach 2,000.