The Saints had to wait until the 52nd minute to make the breakthrough as Austin netted from close range, but it proved to be a comfortable afternoon for the hosts despite a Sam Vokes penalty in the 72nd minute.
Austin's second of the match also came from the penalty spot, and the 27-year-old has said that it is "a dream" to play in a team that create so many opportunities.
"I thought it was going to be one of those afternoons, I missed a few chances, but once we got one we went on to get two or three. I half-celebrated that header in the first half but it was a great save from Tom Heaton," Austin told Sky Sports News.
"We are top-half now and we have got tough fixtures coming up - City away, Chelsea at home and Inter Milan. The international break can hinder you but we have carried on from where we left off. I've got seven goals in the last six but we are seven unbeaten. That's the main thing. We knew it would be a frustrating afternoon if we didn't score early but once we scored one we kicked on.
"This team will always create chances, it's a dream for any striker. I am in the right place at the right time. I don't play for Burnley no more, I have great respect for the club and the fans, Sean Dyche is superb, but I love celebrating goals and I celebrate every one I score. My job is to score goals for Southampton."
Austin, who joined Southampton from Queens Park Rangers in January, has scored seven times in nine matches for the Saints this season.