We study the potential of fully-differential measurements of high-energy dilepton cross-sections at the LHC to probe heavy new physics encapsulated in dimension-6 interaction operators. The assessment is performed in the seven-dimensional parameter space of operators that induce energy-growing corrections to the Standard Model partonic cross-sections at the interference level, and in the two-dimensional subspace associated with the W and Y parameters. A considerable sensitivity improvement is found relative to single-differential measurements, owing to the possibility of probing at the interference level more directions in the seven-dimensional parameter space. The reduction of parton distribution function uncertainties in the fully-differential fit is also found to play a significant role. The results are interpreted in the minimal Z ' new-physics model, providing a concrete illustration of the advantages of the fully-differential analysis. We find that high-energy dilepton measurements can extend the Z ' exclusion and discovery potential well beyond the reach of direct searches in a large region of the parameter space.