This follows from a direct generalization of the Noether normalization lemma.  It is covered in these [notes](http://www.math.lsa.umich.edu/~hochster/615W10/supNoeth.pdf) from Mel Hochster.  These notes prove it in a pretty general form (when the base ring is only an integral domain rather than a field).

Edit: <del>A sufficient condition is that the algebra is finitely generated, but it is clearly not necessary.</del>

Edit 2: I misread the question.  I thought he was asking if A is finitely generated over some polynomial algebra (including infinitely generated polynomial algebras).