We prove that the geodesic flow on a geometrically finite locally symmetric space of negative curvature is exponentially mixing with respect to the Bowen-Margulis-Sullivan measure. The approach is based on constructing a suitable anisotropic Banach space on which the infinitesimal generator of the flow admits an essential spectral gap. A key step in the proof involves estimating certain oscillatory integrals against the Patterson-Sullivan measure. For this purpose, we prove a general result of independent interest asserting that the Fourier transform of measures on $\mathbb{R}^d$ that do not concentrate near proper affine hyperplanes enjoy polynomial decay outside of a sparse set of frequencies. As an intermediate step, we show that the $L^q$-dimension ($1<q\leq \infty$) of iterated self-convolutions of such measures tend towards that of the ambient space. Our analysis also yields that the Laplace transform of the correlation function of smooth observables extends meromorphically to the entire complex plane in the convex cocompact case and to a strip of explicit size beyond the imaginary axis in the case the manifold admits cusps.