Conventional full-waveform inversion (FWI) often fails to retrieve the unknown model parameters from noisy seismic data. A successful FWI implementation usually requires to follow a multistage recovery approach, starting from the retrieval of the lower model wavenumbers (tomography) to those with the higher resolutions (migration). Here, we propose a new method based on the flux-corrected transport (FCT) technique often used in computational fluid dynamics for the removal of instabilities in a shock profile. FCT involves three finite-difference steps: a transport, a diffusion and an antidiffusion process. This third step involves non-linear operators such as maximum and minimum, which are non-differentiable in a classic sense. However, since the seismic source wavelet and the corresponding wavefield are relatively smooth and continuous in nature without any strong ripples like shock waves, we exclude the non-linear step from FCT, which allows us to evaluate the novel FWI gradient efficiently. As a result, we achieve a converging FWI model by gradually reducing the diffusive flux-correction. We demonstrate the versatility of FCT-based FWI on a noisy synthetic data set from the Marmousi II model and a marine field data set from offshore Australia.