Immersed boundary methods are becoming increasingly popular for the computation of unsteady flows around complex geometries using a Cartesian grid. While good results, both qualitative and quantitative, have been obtained, many of the methods rely on low-order corrections to account for the immersed boundary. The objective of the present work is to present a high-order immersed boundary method for the 2-D, unsteady, incompressible Navier-Stokes equations in stream function-vorticity formulation. The method employs an explicit Runge-Kutta (second or fourth order) time integration scheme, fourth-order compact finite-differences for computation of spatial derivatives, and a nine-point, fourth-order compact discretization of the Poisson equation for computation of the stream function. Corrections to the finite-difference stencils are used to maintain high formal accuracy at the immersed boundary, as confirmed by analytical tests. To validate the method in its application to incompressible flows, several physically relevant test cases are computed, including uniform flow past a circular cylinder and Tollmien-Schlichting waves in a boundary layer.