Shifted Chebyshev polynomial functions are used to solve a class of open-loop differential games described by linear state dynamics and quadratic objective functions. Using linear transformation of the variables, the necessary conditions that the open-loop differential games must satisfy are reduced to functional differential equations. The solutions of the functional equations are then obtained in a series of shifted Chebyshev polynomial approximations. The computational procedure is simple and available for computer implementation. It is also shown that the computational results provide satisfactory accuracy compared with those of the exact solutions.