The 21-year-old failed to agree personal terms with Ipswich on Thursday after Swindon had accepted an offer, with Burnley then stepping in.
Austin, who has scored 17 goals for the Robins this season, had asked for a move away from the County Ground.
Neither club have confirmed details of the transfer for the former builder, who had also been linked to Premier League side Blackpool.
Whilst boss at Bournemouth, Burnley manager Eddie Howe almost signed the forward but was prevented by a transfer embargo which was placed on the club at the time.
Speaking before the deal was tied up, Robins boss Danny Wilson told BBC Wiltshire: “You just can’t stand in the way and we have to move on.
“We wish him all the best, he’s done tremendously well in such a short space of time.
“He’s exceeded everybody’s expectations, it has been a real fairytale for him.”
Austin joined Swindon from non-league Poole Town in October 2009.
Since then, he has found the net 37 times in 65 games and signed a new two-and-a-half-year deal in February 2010.