The Hampshire sides had played out an entertaining 2-2 draw at Fratton Park to force a replay, and the second tie was equally absorbing, if less prolific.
An end-to-end contest at the Recreation Ground gave fans of both sides plenty to cheer about.
But it was the home fans celebrating come full time as Mark Molesley bundled home after a goalmouth scramble to help the Conference side eliminate the 2008 cup winners and set up a meeting with Rochdale in round two.
Portsmouth's prized youngster Jed Wallace almost lit up the contest with an impressive first-half volley, but Paul Smith was equal to it in Aldershot's goal.
As the game came to life Molesley saw a diving header drift agonisingly wide of the post before Portsmouth goalkeeper Paul Jones smartly kept Brett Williams' snap shot out.
But the hosts' pressure finally told as a corner from the right was turned goalwards three times, Molesley finally forcing the ball over the line after a string of heroic blocks by Alex Wynter.
