  We establish exponential mixing for the geodesic flow $\varphi_t\colon
T^1S\to T^1S$ of an incomplete, negatively curved surface $S$ with cusp-like
singularities of a prescribed order. As a consequence, we obtain that the
Weil-Petersson flows for the moduli spaces ${\mathcal M}_{1,1}$ and ${\mathcal
M}_{0,4}$ are exponentially mixing, in sharp contrast to the flows for
${\mathcal M}_{g,n}$ with $3g-3+n>1$, which fail to be rapidly mixing. In the
proof, we present a new method of analyzing invariant foliations for hyperbolic
flows with singularities, based on changing the Riemannian metric on the phase
space $T^1S$ and rescaling the flow $\varphi_t$.
