The Euclid space telescope of the European Space Agency (ESA) is designed to provide sensitive and accurate measurements of weak gravitational lensing distortions over wide areas on the sky. Here, we present a weak gravitational lensing analysis of early Euclid observations obtained for the field around the massive galaxy cluster Abell 2390 as part of the Euclid Early Release Observations (ERO) programme. We conducted shape measurements for galaxies down to IE ≲ 26.5 using three independent algorithms (LensMC, KSB+, and SourceXtractor++). Incorporating multi-band photometry from Euclid and Subaru/Suprime-Cam, we estimated photometric redshifts to preferentially select background sources from tomographic redshift bins, for which we calibrated the redshift distributions using the self-organising map approach and data from the Cosmic Evolution Survey (COSMOS). We quantified the residual cluster member contamination and corrected for it in bins of photometric redshift and magnitude using their source density profiles, including corrections for source obscuration and magnification. We reconstructed the cluster mass distribution and jointly fit the tangential reduced shear profiles of the different tomographic bins with spherical Navarro-Frenk-White profile predictions to constrain the cluster mass, finding consistent results for the three shape catalogues and good agreement with earlier measurements. As an important validation test, we compared these joint constraints to mass measurements obtained individually for the different tomographic bins, finding a good level of consistency. More detailed constraints on the cluster properties are presented in a companion paper, which additionally incorporates strong lensing measurements. Our analysis provides a first demonstration of the outstanding capabilities of Euclid for tomographic weak lensing measurements. ★ This paper is published on behalf of the Euclid Consortium.