We report X-ray observations of the newly discovered pulsating white dwarf eRASSU J191213.9−441044 with Spectrum Roentgen Gamma and eROSITA (SRG/eROSITA) and XMM-Newton. The new source was discovered during the first eROSITA all-sky survey at a flux level of f_X(0.2 − 2.3 keV) = 3.3 × 10⁻¹³ erg cm⁻² s⁻¹ and found to be spatially coincident with a G = 17.1 stellar Gaia-source at a distance of 237 pc. The flux dropped to about f_X = 1 × 10⁻¹³ erg cm⁻² s⁻¹ during the three following eROSITA all-sky surveys and remained at this lower level during dedicated XMM-Newton observations performed in September 2022. With XMM-Newton, pulsations with a period of 319 s were found at X-ray and ultraviolet wavelengths occurring simultaneously in time, thus confirming the nature of eRASSU J191213.9−441044 as the second white-dwarf pulsar. The X-ray and UV-pulses correspond to broad optical pulses. Narrow optical pulses that occurred occasionally during simultaneous XMM-Newton/ULTRACAM observations have no X-ray counterpart. The orbital variability of the X-ray signal with a roughly sinusoidal shape was observed with a pulsed fraction of ∼28% and maximum emission at orbital phase ∼0.25. The ultraviolet light curve peaks at around binary phase 0.45. The X-ray spectrum can be described with the sum of a power law spectrum and a thermal component with a mean X-ray luminosity of L_X(0.2 − 10 keV) = 1.4 × 10³⁰ erg s⁻¹. The spectral and variability properties could indicate some residual accretion, in contrast to the case of the prototypical object AR Sco.