One of the most important problems encountered in the thermal management of microelectronics is thermal spreading resistance. This occurs either due to the heat transfer by the conduction mechanism from one solid to another with different cross-sectional areas, or as a result of the heat flow through a conductive solid with a variable cross-sectional area. In this study, both geometric conditions are considered simultaneously. A C++ program code is developed to calculate the thermal spreading resistance in arbitrary curved-edge heat spreaders. A method for automatic numerical generation of a body-fitted curvilinear coordinate system is applied to solve the heat conduction equation on the orthogonal curvilinear grids. A set of Poisson equations is then used to generate two-dimensional grids with grid control along all of the boundaries. The finite difference method is employed to discretize the partial differential equations of the problem. In addition, the Maxwell coordinate system is also used as a special case to demonstrate the similarity of grids generated by numerical and analytical methods. The numerical results of the study are compared with the exact solution, thus illustrating the performance of the approach. Finally, the temperature distribution and thermal spreading resistance are determined for the different shapes of curved edges.