We present results showing an improvement of the accuracy of perturbation theory as applied to cosmological structure formation for a useful range of quasi-linear scales. The Lagrangian theory of gravitational instability of an Einstein-de Sitter dust cosmogony investigated and solved up to the third order in the series of papers, is compared with numerical simulations. In this paper we study the dynamics of pancake models as a first step. the accuracy of several analytical approximations for the modeling of large-scale structure in the mildly non-linear regime was analyzed in the same way, allowing for direct comparison of the accuracy of various approximations. In particular, the " Zel'dovich approximation " (Zel'dovich 1970, 1973, hereafter ZA) as a subclass of the first-order Lagrangian perturbation solutions was found to provide an excellent approximation to the density field in the mildly non-linear regime (i.e. up to a linear r.m.s. density contrast of σ ≈ 2). The performance of ZA in hierarchical clustering models can be greatly improved by truncating the initial power spectrum (smoothing the initial data). We here explore whether this approximation can be further improved with higher-order corrections in the displacement mapping from homogeneity. We study a single pancake model (truncated power-spectrum with power-index n = −1) using cross-correlation statistics employed in previous work. We found that for all statistical methods used the higher-order corrections improve the results obtained for the first-order solution up to the stage when σ(linear theory) ≈ 1. While this improvement can be seen for all spatial scales, later stages retain this feature only above a certain scale which is increasing with time. However, third– order is not much improvement over second–order at any stage. The total breakdown of the perturbation approach is observed at the stage, where σ(linear theory)≈ 2, which corresponds to the onset of hierarchical clustering. This success is found at a considerably higher non-linearity than is usual for perturbation theory. Whether a truncation of the initial power-spectrum in hierarchical models retains this improvement will be analyzed in a forthcoming work.