Kirk Hudson lost possession in midfield and Robert Taylor breezed past two challenges before slotting coolly across the keeper and into the bottom right corner.
In the 39th minute, Hudson atoned for his error when he tucked the ball inside to Louis Soares, who drilled the ball into the bottom right corner to equalise.
Marvin Morgan and Soares missed good chances to put the Shots ahead, but the scores remained all square at half-time.
After the restart, Hudson missed an open goal after Morgan squared across goal. Then John McCombe cleared off the line after Morgan went round Vale keeper Chris Martin as Aldershot continued to press.