In mineral and oil exploration, gravity gradient data can help to delineate small-scale features that cannot be retrieved from gravity measurements. Removing high-frequency noise while preserving the high-frequency real signal is one of the most challenging tasks associated with gravity gradiometry data processing. We present a method to reduce gravity and gravity gradient data noise when both are measured in the same area, based on a least-squares simultaneous inversion of observations and physical constraints, inferred from the gravity gradient tensor definition and its mathematical properties. Instead of handling profiles individually, our noise-reduction method uses simultaneously measured values of the tensor components and of gravity in the whole survey area, benefiting from all available information. Synthetic examples show that more than half of the random noise can be removed from all tensor components and nearly all the noise from the gravity anomaly without altering the high-frequency information. We apply our method to a set of marine gravity gradiometry data acquired by Bell Geospace in the Faroe-Shetland Basin to demonstrate its power to resolve small-scale features.