Context. The TESS team periodically issues a new list of transiting exoplanet candidates based on the analysis of the accumulating light curves obtained by the satellite. The list includes the estimated epochs, periods, and durations of the potential transits. As the point spread function (PSF) of TESS is relatively wide, follow-up photometric observations at higher spatial resolution are required in order to exclude apparent transits that are actually blended background eclipsing binaries (BEBs). Aims. The Gaia space mission, with its growing database of epoch photometry and high angular resolution, enables the production of distinct light curves for all sources included in the TESS PSF, up to the limiting magnitude of Gaia. This paper reports the results of an ongoing Gaia-TESS collaboration that uses the Gaia photometry to facilitate the identification of BEB candidates and even to confirm on-target candidates in some cases. Methods. We inspected the Gaia photometry of the individual sources included in the TESS PSF, searching for periodic dimming events compatible with their ephemerides and uncertainties, as published by TESS. The performance of the search depends mainly on the number of Gaia measurements during transit and their precision. Results. Since February 2021, the collaboration has been able to confirm 126 on-target candidates and exclude 124 as BEBs. Since June 2021, when our search methodology matured, we have been able to identify on the order of 5% as on-target candidates and another 5% as BEBs. Conclusions. This synergistic approach is combining the complementary capabilities of two of the astronomical space missions of NASA and ESA. It serves to optimize the process of detecting new planets by making better use of the resources of the astronomical community.