An algorithm for isometrically embedding curvilinear meshes defined on Riemannian metric spaces into Euclidean spaces of sufficiently high dimension is presented. The method is derived from the Landmark-Isomap algorithm and a previous method for embedding straight-sided meshes. The former is used to decrease the computational complexity of the embedding problem, notably the dense shortest-path problem used to estimate geodesic lengths across the mesh domain as well as the dense eigenvalue decomposition needed to compute the codimension coordinates. A method for defining curvilinear meshes from straight-sided ones in a dimension-independent manner is also discussed. Examples in two- and three-dimensions for both analytic embeddings and analytic metric fields are used to evaluate the method.