In this paper, a numerical method based on polynomial approximation, using Chebyshev polynomial basis, to obtain the approximate solution of generalized pantograph equations with constant coefficients is presented. The technique we have used is an improved Chebyshev matrix method. Some numerical examples, which consist of initial conditions, are given to illustrate the accuracy and efficiency of the method. Also, the results obtained are compared by the known results; the accuracy of solutions and the error analysis are performed.