Both clubs fielded weakened sides because of their league promotion bids.
But two first teamers continued their injury comebacks - Town's Nathan Elder played 90 minutes and Alex Meechan came on in the second half for Telford.
The game appeared to be drifting to a penalty shoot-out when Cork, a first year scholar with the Shrewsbury academy, fired home from inside the box five minutes from time, after being set up by Danny Taylor.
The game attracted only 401 fans to the Greenhous Meadow, as both Graham Turner and Andy Sinton had declared that they would not be using their established first-team players.
Next season the competition will revert to being a pre-season tournament, with The New Saints and Market Drayton Town also invited.
Shrewsbury will play TNS and Telford will face Market Drayton in the semi-finals, on dates to be confirmed.
Source: BBC Sport
Source: BBC Sport