Galaxy cluster gas temperatures (T) play a crucial role in many cosmological and astrophysical studies. However, it has been shown that T measurements can significantly vary between different X-ray telescopes. These T biases can propagate to several cluster applications in which T can be used, such as measuring hydrostatic cluster masses and constraining the angular variation of cosmological parameters. Thus, it is important to accurately cross-calibrate X-ray instruments to account for systematic biases. In this work, we present the cross-calibration between Spectrum Roentgen Gamma/eROSITA (SRG/eROSITA) and Chandra/ACIS and between SRG/eROSITA and XMM-Newton/EPIC using for the first time a large sample of galaxy cluster T. To do so, we used the first eROSITA All-Sky Survey data and the preliminary extremely expanded HIgh FLUx Galaxy Cluster Sample, a large X-ray flux-limited cluster catalog. We spectroscopically measured X-ray T for 186 independent cluster regions with both SRG/eROSITA and Chandra/ACIS in a self-consistent way for three energy bands: 0.7–7 keV (full), 0.5–4 keV (soft), and 1.5–7 keV (hard). We did the same with SRG/eROSITA and XMM-Newton/EPIC for 71 different cluster regions and all three bands. We find that SRG/eROSITA measures systematically lower T than the other two instruments, with hotter clusters deviating more than cooler ones. For the full band, SRG/eROSITA returns 20% and 14% lower T than Chandra/ACIS and XMM-Newton/EPIC, respectively, when the two other instruments each measure kBT ≈ 3 keV. The discrepancy respectively increases to 38% and 32% when Chandra/ACIS and XMM-Newton/EPIC each measure kBT ≈ 10 keV. On the other hand, the discrepancy becomes milder for low-T galaxy groups. Moreover, a broken power law fit demonstrated that there is a break at the SRG/eROSITA-Chandra/ACIS scaling relation at kBT ≈ 1.7 − 2.7 keV, depending on the energy band. The soft band shows a marginally lower discrepancy compared to the full band. In the hard band, the cross-calibration of SRG/eROSITA and the other instruments show very strong differences. We tested several possible systematic biases (such as multiphase cluster gas, Galactic absorption, non-Gaussian scatter, and selection effects) to identify the reason behind the cross-calibration discrepancies, but none could significantly alleviate the tension. For now, it is most likely that the systematically lower SRG/eROSITA T can be attributed to systematic effective area calibration uncertainties; however, the exact role of multiphase cluster gas in the observed T discrepancies needs to be further investigated. Furthermore, we provide conversion factors between SRG/eROSITA, Chandra/ACIS, and XMM-Newton/EPIC T that will be beneficial for future cluster studies that combine SRG/eROSITA T with data from other X-ray instruments. Finally, we also provide conversion functions between the official eRASS1 cluster catalog T and the equivalent core and core-excised Chandra/ACIS and XMM-Newton/EPIC T.