We give a new and very concise proof of the existence of a holomorphic continuation for a large class of twisted multivariable zeta functions. To do this, we use a simple method of “decalage” that avoids using an integral representation of the zeta function. This allows us to derive explicit recurrence relations between the values at T–tuples of negative integers. This also extends some earlier results of several authors where the underlying polynomials were products of linear forms.