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The strong mixing of close levels with two valence electrons in Be-like xenon greatly complicates ab initio QED calculations beyond the first-order approximation. Due to a strong interplay between the electron-electron correlation and QED effects, the standard single-level perturbative QED approach may fail, even if it takes into account the second-order screened QED diagrams. In the present Letter, the corresponding obstacles are overcome by working out the QED perturbation theory for quasidegenerate states. The contributions of all the Feynman diagrams up to the second order are taken into account. The many-electron QED effects are rigorously evaluated in the framework of the extended Furry picture to all orders in the nuclear-strength parameter $αZ$. The higher-order electron-correlation effects are considered within the Breit approximation. The nuclear recoil effect is accounted for as well. The developed approach is applied to high-precision QED calculations of the ground and singly excited energy levels in Be-like xenon. The most accurate theoretical predictions for the binding and excitation energies are obtained. These results deviate from the most precise experimental value by $3σ$ but perfectly agree with a more recent measurement.