The Tigers (3-9, 2-4 ODAC) got all of the offense they would need with a Bo Burns to James Lawrence goal in the 12th minute.
Shenandoah (2-10-1, 0-5) nearly matched H-SC in shots, taking nine of the game's 19, but saw the Tigers pull away with goals in the 43rd, 63rd and 73rd minutes.
The starting goalkeepers James Dutcher for SU and Blake Carrey for Hampden-Sydney, split eight saves.
Shenandoah also had a 6-4 advantage in corner kicks.
Robert Stephens led the victors with five points on two goals and one assist. He figured in the final three goals of the contest.
SU is back on the pitch on Wednesday at Roanoke in another ODAC matchup.