The Eagles netted the equalizer with just under eight minutes to play in regulation before getting the game-winner 10 minutes into the first sudden-victory OT.
The overtime loss is the fourth this season for the Hornets and the third in six ODAC contests for the team.
Bridgewater outshot SU 17-7 for the game, including a combined 9-2 in the second half and OT.
The Eagles also owned a 14-6 advantage in penalty corners.
Shenandoah goalkeeper Sarah Clare had six saves.
SU returns to action on Wednesday evening with a 4:00 home ODAC contest against nationally ranked Lynchburg College.