Fringe projection techniques are popular tools for three-dimensional shape measurement in optical metrology. In these techniques, the information about the shape or the surface profile of an object is usually encoded in the phase of a recorded fringe pattern. The paper presents a high-order instantaneous moments based method for phase estimation in fringe projection. The method employs a piecewise polynomial phase approximation approach and uses high-order instantaneous moments to determine the polynomial phase coefficients to estimate the phase. The method's working is demonstrated using simulation examples and its practical applicability is validated through a shape measurement experiment.