We introduce a new numerical method to simulate geophysical marine controlled source electromagnetic (CSEM) measurements for the case of 2D structures and finite 3D sources of electromagnetic (EM) excitation. The method of solution is based on a spatial discretization that combines a 1D Fourier transform with a 2D self-adaptive, goal-oriented, hp-Finite element method. It enables fast and accurate simulations for a variety of important, challenging and practical cases of marine CSEM acquisition. Numerical results confirm the high accuracy of the method as well as some of the main physical properties of marine CSEM measurements such as high measurement sensitivity to oil-bearing layers in the subsurface. In our model, numerical results indicate that measurements could be affected by the finite oil-bearing layer by as much as 104% (relative difference). While the emphasis of this paper is on EM simulations, the method can be used to simulate different physical phenomena such as seismic measurements.