In this article, we propose new spectral solutions for odd-order two-point boundary value problems. A numerical algorithm based on the use of collocation methods is implemented and analyzed. A new matrix of derivatives of certain Legendre basis polynomials is derived to serve in deriving the proposed algorithm. Our algorithm has the advantage that it is applicable to both linear and nonlinear odd-order two-point boundary value problems. The convergence and error analysis of the suggested method are investigated. Five illustrative examples supported with comparisons are displayed to ensure the efficiency and applicability of the proposed algorithm.