The former Swindon striker started his first game since the end of February following surgery to repair a shoulder problem.
Austin claimed the Clarets' opening goal after just seven minutes at Gigg Lane while rookie frontman Alex MacDonald and Chris Iwelumo were also on target and Chris Eagles missed a penalty.
Boss Eddie Howe, who saw MacDonald and Iwelumo score for the second successive game, said: "You always want your strikers to have a good pre-season and score goals and so I'm delighted for all of them.
"Charlie in particular will get huge confidence from that. I have never doubted his ability. Charlie is a goalscorer and a fantastic finisher who has been champing at the bit."