Austin was in the lineup at first base and batting seventh Thursday night as New York tried for a four-game sweep of its Subway Series against the crosstown-rival Mets. Chase Headley was given a rest.
With projected regular Greg Bird sidelined by an ankle injury most of the year, the Yankees have started nine players at first base this season.
Cooper was acquired in a trade with Milwaukee on July 13 to help shore up the position. He is batting .326 with six RBIs in 13 games for New York.