Saints had the better of the first half at The County Ground on Saturday, with centre-back Aaron Martin hitting the crossbar with a header from a Jason Puncheon corner.
David Connolly also came close, his shot saved sharply by Phil Smith in the Swindon goal.
But Swindon did have their chances in the opening 45 minutes, Austin's low half-volley being brilliantly diverted wide by Kelvin Davis while Vincent Pericard's excellent header was ruled out for offside.
The second half started in similar fashion to that of the first half, both sides having chances but failing to convert - the visitors hitting the bar for a second time through Dan Harding.
But six minutes later Austin volleyed home from six yards to grab what turned out to be game's only goal, although the day ended badly for the young striker when he was later stretchered off with what appeared to be an arm injury.