The aim of this paper is to present an improvement of the incremental Newton method proposed by Iannazzo [SIAM J. Matrix Anal. Appl., 28:2 (2006), 503–523] for approximating the principal $p$-th root of a matrix. We construct and analyze an incremental Chebyshev method with better numerical behavior. We present a convergence and numerical analysis of the method, where we compare it with the corresponding incremental Newton method. The new method has order of convergence three and is stable and more efficient than the incremental Newton method.