Austin was playing non-league football for Poole Town at the start of October but Danny Wilson's gamble on the 20-year-old has paid dividends in a big way as he headed home his 20th of the season to cap a man-of-the-match display.
Danny Ward grabbed a quick second for Town on the hour mark to ensure the pair capitalised on a dominant first-half display in which they missed three great openings between them.
But Deon Burton put his head where it hurts in the 65th minute to ensure Charlton, who also hit a post through skipper Nicky Bailey, left the encounter finely poised heading into Monday's second leg at The Valley.
Credit must go to Swindon who, bidding to reach the second tier for the first time in 10 years, also hit the woodwork through Vincent Pericard but emerged triumphant without the services of 26-goal leading scorer Billy Paynter as he was hit with a late hamstring injury.
Amid an electric atmosphere at a sell-out County Ground, Austin was denied an opener with just two minutes on the clock as Addicks goalkeeper Darren Randolph did well to palm away his free header.
Randolph was certainly the busier of the two keepers in the first half and was called upon again to turn Austin's header for a corner after 12 minutes.
Bailey dragged the visitors' first chance wide but they posed little threat in the opening 45 minutes.
Experienced midfielder Jonathan Douglas, a key member of the Robins' play-off charge, was next to head wide just before the half-hour mark.
And having already seen Austin fail to convert two great openings, Ward should have finally broken the deadlock for the hosts shortly after.
Having been sent clear by Alan Sheehan, the on-loan Bolton forward could only fire across the face of Randolph's goal and inches wide of his left-hand post. Addicks boss Phil Parkinson must have felt fortunate to still be on level terms at the interval and the influential Bailey almost made Swindon pay for their wastefulness four minutes after the restart.
The ball worked its way to the talented midfielder on the left, he side-stepped Kevin Amankwaah and curled a delightful effort towards the top corner that had David Lucas beaten but rebounded away off the inside of a post.
And just three minutes later Swindon - and Austin - finally got the goal they deserved.
Charlton failed to learn their lessons from the first half and the frontman peeled off his marker with ease to plant Sheehan's left-wing cross past a helpless Randolph.
Ward was next to make up for his first-half profligacy to double the home side's lead, drilling low past Randolph after Simon Ferry had taken advantage of a quick break and a two-on-one situation.
The encounter had burst into life in the second half and Burton reduced Charlton's arrears soon after, bravely heading home Randolph's long ball forward as Lucas came but failed to collect.
And the Addicks number 10 somehow failed to head level from a matter of yards just a few minutes later.
Both sides had half chances in the final quarter to change the pattern of the game, with Pericard heading against the woodwork for a Swindon side who will take a small lead to London for Monday's return leg.