We consider the problem of locating point sources in the planar domain from overdetermined boundary measurements of solutions of Poisson's equation. In this paper, we propose a novel technique, termed "analytic sensing," which combines the application of Green's theorem to functions with vanishing Laplacian-known as the "reciprocity gap" principle-with the careful selection of analytic functions that "sense" the manifestation of the sources in order to determine their positions and intensities. Using this formalism we express the problem at hand as a generalized sampling problem, where the signal to be reconstructed is the source distribution. To determine the positions of the sources, which is a nonlinear problem, we extend the annihilating-filter method, which reduces the problem to solving a linear system of equations for a polynomial whose roots are the positions of the point sources. Once these positions are found, resolving the according intensities boils down to solving a linear system of equations. We demonstrate the performance of our technique in the presence of noise by comparing the achieved accuracy with the theoretical lower bound provided by Cramer-Rao theory.