Aldershot, one of the few part-time teams still left in the Coference, struck first when Aaron McLean finished clinically from a Ray Warburton head-down.
It looked like they could regain the Football League status they lost after going bust in 1992.
The Shrews equalised shortly before the break when Duane Darby fired an vicious shot high into the net.
Both sides tired towards the end of normal time, with a sliced clearance from Aldershot defender Warburton almost gifting Shrewsbury a late winner.
Shrewsbury's star striker Lee Rodgers fluffed the opening penalty of the shootout, but was rescued by goalie Scott Howie who kept out Aldershot's opening effort.
TV replays suggested he had taken two steps forward before diving, something he looked to do again as he pulled off a further two saves to seal the shootout win 3-0 for the Shrews.
They join Conference champions Chester City in returning to the Third Division next season.