Analytical solutions are presented for eigenvalues, eigenfunctions of D-dimensional Schrodinger equation having Eckart potential within Nikiforov-Uvarov method. This uses a new, improved approximation for centrifugal term, from a combination of Greene-Aldrich and Pekeris approximations. Solutions are obtained in terms of hypergeometric functions. It facilitates an accurate representation in entire domain. Its validity is illustrated for energies in an arbitrary ℓ≠0 quantum state. Results are compared for a chosen set of potential parameters in different dimensions. In short, a simple accurate approximation is offered for Eckart and other potentials in quantum mechanics, in higher dimension.