A novel numerical method to solve asymmetric adhesive contact problems in rectangular coordinates has been developed. Surface interaction is modelled using an interface potential, deformation is coupled using Green's functions for a half space, and the resulting system of equations is solved by a relaxation technique. The method can handle arbitrary surface topography and properties. Compared with previous methods, this numerical scheme is much easier to implement and is just as accurate. Here, it is applied to two adhesive contact problems: one between a sphere and a cylinder; and the other between two identical cylinders in oblique contact. The numerical results reveal inaccuracies in elliptical contact theory when the skew angles between the two cylinders are small and the resulting contact is highly eccentric. The pull-off forces show an indiscernible decrease with decreasing value of the skew angle, which is quite different from the elliptical JKR theory. This technique can be used to solve adhesive contact problems that involve partial contact or complex geometry, such as rippled or rough surfaces.