In this paper, the homotopy perturbation method (HPM) is developed to solve the focusing Manakov system of coupled nonlinear Schrödinger equations in one space variable and two space variables. The advantages of this method are, fast convergent, does not require discretizations of space-time variables and does not require to solve the resultant nonlinear system of discrete equations. Numerical examples are shown to demonstrate the accuracy and capability of the method. The accuracy of the method is verified by ensuring that the conserved quantities remain almost constant.