Understanding the formation and evolution of the cosmic web of galaxies is a fundamental goal of both theoretical and observational cosmology, using various tracers of the cosmic large-scale structure at an ever wider range of redshifts. Our principal aim is to advance the mapping of the cosmic web at high redshifts using observational and synthetic catalogues of quasars (QSOs), which offer a powerful probe of structure formation and the validity of the concordance cosmological model at the largest scales in the Universe. In this analysis, we selected 708483 quasars at 0.8<z<2.2 from the Quaia data set, allowing an extended reconstruction of the matter density field using 24372deg^2^ sky area with a well-understood selection function, and thus going beyond the capacity of previous studies. Using the REVOLVER method, we created catalogues of voids and clusters based on the estimation of the local density at QSO positions with Voronoi tessellation. We tested the consistency of Quaia data and 50 realistic mock catalogues, including various parameters of the voids and clusters in characteristic subsets of the data, and also measurements of the density profiles of these cosmic super-structures at R=~100h^-1^Mpc scales. We identified 12820 voids and 41154 clusters in the distribution of Quaia quasars. We found ~5-10% level agreement between data and the ensemble of the 50 mocks considering void and cluster radii, average inner density, and density profiles at all redshifts. In particular, we tested the role of survey mask proximity effects in the void and cluster detection, which although present in the data, are consistent in simulations and observations. Testing the extremes, the largest voids and clusters reach Reff=~250h^-1^Mpc and Reff=~150h^-1^Mpc, respectively, but without evidence for ultra-large cosmic structures exceeding the dimensions of the largest structures in our mock catalogues. Our data-analysis results highlight the capacity of Quaia quasars to robustly map the high-z cosmic web, further supported by the fully consistent statistical results from 50 mock catalogues. As an important deliverable, we share our density field estimation, void catalogues, and cluster catalogues with the public, allowing various additional cross-correlation probes in the high-z cosmic web