Offshore ocean bottom pressure gauges (OBPs) are often used to estimate the spatial distribution of the initial sea-surface height associated with offshore earthquakes (the tsunami source model). However, the sensors sometimes record pressure changes that are neither related to tsunamis nor seafloor coseismic displacements (the non-tsunami components) due to sensor rotation or tilt associated with ground shaking or due to long-term mechanical drift. These non-tsunami components can be a source of error when accurately estimating the tsunami source model and thus need to be removed to provide reliable coastal tsunami forecasts. This paper proposes a new method that uses time-derivative waveforms of the pressure time-series from OBP records to robustly estimate the tsunami source model, even when OBP data are perturbed by non-tsunami components. Using OBP data associated with the 2011 Off-Miyagi earthquake (Mw 7.2) and the 2016 Off-Mie earthquake (Mw 5.9), the performance of the method was evaluated when reducing artefacts due to non-tsunami components. The tsunami source model was found to be largely distorted when a conventional inversion method was used (because of the non-tsunami components). However, the artefact was dramatically reduced when using time-derivative waveforms, and the predicted coastal tsunami waveforms fitted reasonably with those of observations, thereby suggesting that the new method effectively suppresses artefacts caused by non-tsunami components. As the tsunami source models estimated from pressure and time-derivative waveforms should be similar when OBP data are not perturbed by non-tsunami components, we would be able to assess whether OBP data are perturbed by non-tsunami components by evaluating that the tsunami source models estimated from pressure waveforms and from time-derivative waveforms are similar to each other.