We present the first X-ray detections of ultracool dwarfs (UCDs) from the first all-sky survey of the extended ROentgen Survey with an Imaging Telescope Array (eROSITA) onboard the Russian-German Spektrum-Roentgen-Gamma mission, SRG. We use three publicly available input catalogs of spectroscopically confirmed UCDs and Gaia-selected UCD candidates that together comprise nearly 20000 objects. In a careful source identification procedure we first extract all X-ray sources from the catalog of the first survey, eRASS1, that have an UCD or candidate within three times their positional uncertainty. Then we examine all Gaia objects in the vicinity of these 96 X-ray sources. We associate the most plausible counterpart on the basis of their spatial separation to the X-ray position and their multi-wavelength properties. This way we find 40 UCDs that have a secure identification with an X-ray source (i.e. bona-fide UCD X-ray emitters) and 18 plausible UCD X-ray emitters for which we consider it likely that the X-ray source has its origin in the UCD. Twentyone of the bona-fide and plausible X-ray emitting UCDs have a spectroscopic confirmation, while the others have been selected based on Gaia photometry and we compute spectral types from the G-J color. The spectral types of the X-ray emitting UCDs and candidates range between M5 and M9. The distances of the eRASS1 UCDs range from 3.5 to 190 pc. The spectroscopically confirmed UCDs at the high end of the distance distribution are known to be members of nearby star forming regions. The majority of the UCDs from the eRASS1 sample show a ratio of X-ray to bolometric luminosity well above the canonical saturation limit of log(Lx/Lbol)~-3. For the two most extreme outliers we have verified the hypothesis that these high values are due to flaring activity through an analysis of the eRASS1 lightcurve. X-ray spectra could be analysed for the two brightest objects in terms of count rate, showing both an emission-measure weighted plasma temperature of <kT>=0.75keV. These observations demonstrate the potential of eROSITA for advancing our knowledge on the faint coronal X-ray emission from UCDs by building statistical samples for which the average X-ray brightness, flares and coronal temperatures can be derived.