An efficient higher-order method of moments (MoM) solution of volume integral equations is presented. The higher-order MoM solution is based on higher-order hierarchical Legendre basis functions and higher-order geometry modeling. An unstructured mesh composed of 8-node trilinear and/or curved 27-node hexahedral elements is used for accurate representation of the scattering dielectric object. The permittivity of the object is allowed to vary continuously as a function of position inside each element. It is shown that the condition number of the resulting MoM matrix is reduced by several orders of magnitude in comparison to existing higher-order hierarchical basis functions. Consequently, an iterative solver can be applied even for high expansion orders. Numerical results demonstrate excellent agreement with the analytical Mie series solution for a dielectric sphere as well as with results obtained by other numerical methods.