Second-half strikes by centre-backs John Brough and Greg Heald made it 2-0, with Ahmed Deen netting his first for the Hampshire club for the third. Matty Miller struck a consolation goal for the Blues.
Burnham were making their first ever appearance at this stage of the FA Cup, after securing another Conference National scalp in the last qualifying round by beating Canvey Island after a replay at The Gore.
A late Adam Logie penalty set up the tie with the Shots. Aldershot got past Bromley in the previous round and were looking to put their indifferent league form aside by going for the victory.
Shots boss Terry Brown was a relieved man at the final whistle, as Tim Sills, his only recognised striker, was unable to appear for the second half, suffering with a shoulder injury.
Brown later admitted he was wondering where his goals would come from. Aldershot should have taken the lead on 18 minutes when the impressive Omari Coleman was denied by a determined Jon Horsted goal-line clearance.
Burnham went close on 20 minutes, but Daryl Jones' free-kick onto Anthony Alleyne's head was saved by Nikki Bull.
Aldershot went ahead on 53 minutes. Darren Barnard's 20-yard free-kick was only blocked by Jamie Jackson, but the alert Brough found the net.
The Shots doubled their lead on 71 minutes through Heald. Nick Crittenden crossed from the right and Heald's precise header sailed past Jackson.
With the game stretched, Aldershot's better fitness levels allowed them to take a grip of the game and they duly scored again. Deen lashed home after Crittenden and Craig Holloway combined from the right. Burnham for all their effort were gutsy, but despite Miller's late strike were outplayed.