The study of the dense gluonic matter emerging in QCD at high energies represents one of the main goals of the experimental program at the future Electron-Ion Collider. We demonstrate that dijet production via inelastic diffraction is a promising channel for probing gluon saturation, including in the hard regime at high photon virtuality, where the collinear factorisation applies. By ``inelastic diffraction'' we mean a process in which the two hard jets -- a quark-antiquark pair generated by the decay of the virtual photon -- are accompanied by a softer gluon jet, emitted by the quark or the antiquark. This process can be described as the elastic scattering between an effective gluon-gluon dipole and the nucleus. The cross section takes a factorised form, between a hard factor and a unintegrated (``Pomeron'') gluon distribution describing the transverse momentum imbalance between the hard dijets. The dominant contribution comes from the black disk limit and leads to a dijet imbalance of the order of the target saturation momentum $Q_s$ evaluated at the rapidity gap. Integrating out the dijet imbalance, we obtain a collinear factorisation where the initial condition for the DGLAP evolution is set by gluon saturation.