Sparta took a third minute lead through Johan Voskamp who beat the offside trap before slotting past Tom Heaton.
But the Clarets roared into action with three goals in 19 minutes. Austin's first came in the fifth when he got on the end of Danny Lafferty's cross to plant his header past the Rotterdam goalkeeper.
Austin missed when clean through moments later but he made amends after 22 minutes when he crossed for Dean Marney to slam home from six yards.
Danny Ings got onto the scoresheet two minutes later when he neatly exchanged passes with Austin before coolly slotting home.
And Austin grabbed his second 18 minutes from time after getting on the end of a David Jones shot.