A new statistical technique is developed for physics parameter estimation at the Large Hadron Collider (LHC) that uses modern deep-learning tools to realize a more general and fundamental approach to data analysis compared to the ad-hoc techniques commonly used. Coming under the general umbrella of Neural Simulation-Based Inference (NSBI) techniques, the new workflow uses a large number of Neural Networks (NNs) to directly learn event-by-event likelihood ratios and thus handles high-dimensional parameter estimation without the need to bin data into low-dimensional summary histograms. We developed novel techniques for parameterizing the likelihood ratios as a function of a large number of parameters common in LHC experiments and created modern computational workflows that make it possible to apply NSBI to a full-scale ATLAS experiment analysis. A measurement of the Higgs boson in the off-shell phase space is then performed using the new workflow, in the $H^*\to ZZ \to 4\ell$ channel. The evidence sensitivity is increased by a factor of $3.1$ using the new technique in the $H^*\to ZZ \to 4\ell$ channel, compared to the previous measurement published by ATLAS using the same Run-2 data with more standard techniques. A combination of this channel with the off-shell $H^*\to ZZ\to 2\ell2\nu$ channel is done finding evidence for the off-shell Higgs boson with $3.7 \sigma$ confidence level, superseding the last measurement. The off-shell measurement is then combined with the on-shell measurement for an indirect measurement of the Higgs boson decay width under a few assumptions. The observed (expected) value of the Higgs boson width at 68\% CL is $4.3^{+2.7}_{-1.9}$ ($4.1^{+3.5}_{-3.4}$)~MeV. This new method is promising for a wide range of measurements at the LHC, where no single observable may be optimal to scan over the entire theoretical phase space under consideration, or where binning data into histograms could result in a loss of sensitivity. It also offers easy re-interpretability and broader use.