A new fully non-linear reconstruction algorithm for the accurate recovery of the Baryonic Acoustic Oscillations (BAO) scale in two-point correlation functions is proposed, based on the least-action principle and extending the Fast Action Minimisation method by Nusser & Branchini (2000). Especially designed for massive spectroscopic surveys, it is tested on dark-matter halo catalogues extracted from the deus-fur ΛCDM simulation to trace the trajectories of up to ∼ 207, 000 haloes backward-in-time, well beyond the first-order Lagrangian approximation. The new algorithm successfully recovers the BAO feature in real and redshift-space in both the monopole and the anisotropic two-point correlation function, also for anomalous samples showing misplaced or absent signature of BAO. In redshift space, the non-linear displacement parameter Σ NL is reduced from 11.8±0.3h −1 Mpc at redshift z = 0 to 4.0±0.5h −1 Mpc at z 37 after reconstruction. A comparison with the first-order Lagrangian reconstruction is presented, showing that this techniques outperforms the linear approximation in recovering an unbiased measurement of the acoustic scale.