The win was the Chippewas' seventh straight.
Brown rushed 4 yards for a touchdown, then caught a 29-yard scoring pass from LeFevour with 9:16 left in the first half to put Central Michigan (7-1, 5-0 Mid-American Conference) ahead 14-7.
Matt Norsic kicked a 40-yard field goal to bring Bowling Green (3-5, 2-2) within 14-10 at halftime.
After Andrew Aguila made a 38-yard field goal early in the third quarter, LeFevour threw a 9-yard touchdown pass to Bryan Anderson with 10:02 remaining.
LeFevour was also Central Michigan's leading rusher with 128 yards on 22 carries.
Tyler Sheehan was 35-for-55 passing for 341 yards with one touchdown and one interception for the Falcons, who were held to 20 yards rushing.