Context. One key ingredient in using galaxy clusters as a precision cosmological probe in large X-ray surveys is understanding selection effects. The dependence of the X-ray emission on the square of the gas density leads to a predominant role of cool cores in the detection of galaxy clusters. The contribution of cool cores to the X-ray luminosity does not scale with cluster mass and cosmology and therefore affects the use of X-ray clusters in producing cosmological constraints. Aims. One of the main science goals of the extended ROentgen Survey with an Imaging Telescope Array (eROSITA) mission is to constrain cosmology with a wide X-ray survey. We propose an eROSITA galaxy cluster detection scheme that avoids the use of X-ray cluster centers in detection. We calculate theoretical expectations and characterize the performance of this scheme by simulations. Methods. We performed Monte Carlo simulations of the upcoming eROSITA mission, including known foreground and background components. By performing realistic simulations of point sources in survey mode, we searched for spatial scales where the extended signal is not contaminated by the point-source flux. We derive a combination of scales and thresholds, which result in a clean extended source catalog. We designed the output of the cluster detection, which enables calibrating the core-excised luminosity using external mass measurements. We provide a way to incorporate the results of this calibration in producing the final core-excised luminosity. Results. Similarly to other galaxy cluster detection pipelines, we sample the detection space of the flux – cluster core radius of our method and find many similarities with the pipeline used in the 400d survey. Both detection methods require large statistics on compact clusters in order to reduce the contamination from point sources. The benefit of our pipeline consists of the sensitivity to the outer cluster shapes, which are characterized by large core sizes with little cluster to cluster variation at a fixed total mass of the cluster. Conclusions. Galaxy cluster detection through cluster outskirts improves the cluster characterization using eROSITA survey data and is expected to yield well-characterized cluster catalogs with simple selection functions.