The dynamic range of imaging detectors flown on-board X-ray observatories often only covers a limited flux range of extrasolar X-ray sources. The analysis of bright X-ray sources is complicated by so-called pile-up, which results from high incident photon flux. This nonlinear effect distorts the measured spectrum, resulting in biases in the inferred physical parameters, and can even lead to a complete signal loss in extreme cases. Piled-up data are commonly discarded due to resulting intractability of the likelihood. As a result, a large number of archival observations remain underexplored. We present a machine learning solution to this problem, using a simulation-based inference framework that allows us to estimate posterior distributions of physical source parameters from piled-up eROSITA data. We show that a normalizing flow produces better-constrained posterior densities than traditional mitigation techniques, as more data can be leveraged. We consider model- and calibration-dependent uncertainties and the applicability of such an algorithm to real data in the eROSITA archive.