They were watched by a massive 8,185 attendance - the fourth biggest Conference gate of all time.
However, the U's made hard work of beating a spirited Shots side, who probably had the better chances until Burgess converted a 73rd minute penalty.
Yemi Odubade won the spot kick with a burst of pace, cutting into the box from the right. Before he had the chance to shoot, Curtis Osanu brought him down. Osanu was yellow-carded, as was Shots goalkeeper Nikki Bull for protesting at the decision.
Oxford's regular penalty-taker, Rob Duffy, had just been substituted, so Burgess stepped up - and confidently fired into the top right corner.
Then six minutes from time, a bad mistake by Rhys Day let in Burgess for a run at goal. He tried to go around the midfielder, who robbed him and then raced forward. With two supporting players for choice should he need it, Burgess drew the keeper and planted a shot into the net from the edge of the box.
United weren't at their best, and struggled to open up the Aldershot defence at times, relying heavily on crosses from both flanks.
But their huge crowd went home more than happy . . . singing at the end "we are invincible".
Jim Smith's men played with a lot of width. Rufus Brevett delivered a succession of top-drawer crosses from the left while Eddie Anaclet was full of running and trickery on the right.
It was from one of Brevett's crosses behind the Shots defence that Chris Hargreaves beat Bull, only to see his shot smash against the bar.
Burgess also went close after a slick three-man move from the home side, but the midfielder slightly sliced his 25-yard drive which arced wide.
With their own fans in good voice, Aldershot had plenty of attacking threat and they had the better of the first-half chances.
Only Ryan Williams will know how he didn't score just before the break when David Lee picked him out with a cross from the right. He arrived unnoticed at the back past but scooped his simple close-range volley wide.
Before that the skilful Joel Grant had a powerful 18-yard drive parried by keeper Billy Turley, and he should have done better with a free header which he couldn't guide goalwards.
The Shots were dealt a blow when Dave Winfield hobbled off injured midway through the first half, Rhys Day taking over at centre half.
Jim Smith had sprung a surprise by selecting Odubade ahead of Steve Basham but although Odubade saw plenty of the ball, there was little of the usual link-up between him and Rob Duffy.
Odubade did manage an effort on goal with a glancing header from yet another Brevett centre, but it flashed wide.
Aldershot made a bright start to the second period with Curtis Osanu seeing his 25-yard blast arrowing just past the top right corner of Turley's goal, and then only a great saving tackle halting Mark Molesley after he had jinked past Barry Quinn.
The visitors seemed to be enjoying themselves as Dean Smith then forced Turley to turn his low shot around a post.
It was a sign of Oxford's frustration that they weren't having things their own way that Duffy was substituted on 66 minutes.
And within seconds of Basham arriving on the pitch, he cushioned a header for Odubade, whose first touch let him down, and the chance was gone. But Odubade proved too nimble for Osanu as he cut into the box from the right in the 73rd minute, and that proved the turning point of the match.