The game had looked to have been heading for stalemate until the striker poked the ball home six minutes from time following a goalmouth scramble after his initial effort had been partially cleared by the Chester defence.
Aldershot could claim to have deserved all three points having struck the woodwork twice in the course of the game.
First John Grant was unfortunate when his powerfully struck shot from outside the box in the 26th minute cannoned back off the woodwork.
And ten minutes later, the visitors found themselves cursing their luck again when Louis Soares struck a right-foot shot from near the penalty spot which hit the post.
Chester's best chance in the first half came on 32 minutes when a clearance from Paul Linwood fortuitously sent Eddie Johnson in on goal only for the striker's touch to take him wide and he opted to try and find Ryan Lowe rather than shoot meaning the chance was gone.
Five minutes into the second half, the home side wasted a glorious opportunity when a corner from Lowe picked out Kevin Roberts, but the defender headed marginally of the target.
Morgan headed a decent chance just past the post and at the other end Lowe was wasteful, hitting a good opportunity well wide of the target after he had been fed by Jay Harris.
The visitors did have the ball in the net on 78 minutes only for Grant's header to be ruled out for offside, but it did not matter as Morgan bundled the winner home.