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Theoretical interpretations of particle physics data, such as the determination of the Wilson coefficients of the Standard Model Effective Field Theory (SMEFT), often involve the inference of multiple parameters from a global dataset. Optimizing such interpretations requires the identification of observables that exhibit the highest possible sensitivity to the underlying theory parameters. In this work we develop a flexible open source framework, ML4EFT, enabling the integration of unbinned multivariate observables into global SMEFT fits. As compared to traditional measurements, such observables enhance the sensitivity to the theory parameters by preventing the information loss incurred when binning in a subset of final-state kinematic variables. Our strategy combines machine learning regression and classification techniques to parameterize high-dimensional likelihood ratios, using the Monte Carlo replica method to estimate and propagate methodological uncertainties. As a proof of concept we construct unbinned multivariate observables for top-quark pair and Higgs+$Z$ production at the LHC, demonstrate their impact on the SMEFT parameter space as compared to binned measurements, and study the improved constraints associated to multivariate inputs. Since the number of neural networks to be trained scales quadratically with the number of parameters and can be fully parallelized, the ML4EFT framework is well-suited to construct unbinned multivariate observables which depend on up to tens of EFT coefficients, as required in global fits.