  Given an associative ring $A$, we present a new approach for establishing the
finiteness of the big finitistic projective dimension $\operatorname{FPD}(A)$.
The idea is to find a sufficiently nice non-positively graded differential
graded ring $B$ such that $\mathrm{H}^0(B) = A$ and such that
$\operatorname{FPD}(B) < \infty$. We show that one can always find such a $B$
provided that $A$ is noetherian and has a noncommutative dualizing complex. We
then use the intimate relation between $\operatorname{\mathsf{D}}(B)$ and
$\operatorname{\mathsf{D}}(\mathrm{H}^0(B))$ to deduce results about
$\operatorname{FPD}(A)$. As an application, we generalize a recent sufficient
condition of Rickard, for $\operatorname{FPD}(A) < \infty$ in terms of
generation of $\operatorname{\mathsf{D}}(A)$ from finite dimensional algebras
over a field to all noetherian rings which admit a dualizing complex.
