The triumph is the 11th straight for Shenandoah (7-11, 4-7 USA South) over AU (10-8, 6-5) in Winchester.
With today's game being the final league matchup between the two schools, the Hornets complete the long-time USA South series with a 28-14 advantage.
After the two teams traded the lead seven times and were tied on two other occasions in the opening nine minutes of the game, Shenandoah took control with a 10-0 run from the 10:06 to 6:29 marks of the first.
Sam Akers gave SU the lead for good with a three-point play off of a fastbreak at 8:16 and Averett went cold from the field.
The Cougars, who shot 24 percent (13 for 55) in the contest, went from the 10:45 mark of the first to the 16:45 mark of the second without scoring a field goal.
Shenandoah trailed by three points when this drought started and had a 12-point lead when Liz Martin finally snapped it with a three-point play.
The Hornets remained in control of the contest, however, never allowing their lead to slip below nine points the rest of the way.
Akers completed the SU players in double-figures with 12 points.
Martin had game-highs in points, 17, and rebounds, 12, for Averett.
Shenandoah had a 43-39 advantage on the glass and posted 28 points in the paint to the Cougars 20.
The Hornets also scored all 19 points registered by bench players in the contest.
Shenandoah hosts Mary Baldwin on Tuesday in a 7 p.m. USA South Conference contest.