Three-dimensional (3D) tissue culture has attracted a great deal of attention as a result of the need to replace the conventional two-dimensional cell cultures with more meaningful methods, especially for understanding the sophisticated nature of native tumor microenvironments. However, most techniques for 3D tissue culture are laborious, expensive, and limited to spheroid formation. In this study, a low-cost and highly effective nanofibrous scaffold is presented for spontaneous formation of reproducible 3D breast cancer microtissues. Experimentally, aligned and non-aligned chitosan/poly(ethylene oxide) nanofibrous scaffolds were prepared at one of two chitosan concentrations (2 and 4 wt %) and various electrospinning parameters. The resulting fabricated scaffolds (C2P1 and C4P1) were structurally and morphologically characterized, as well as analyzed in silico. The obtained data suggest that the fiber diameter, surface roughness, and scaffold wettability are tunable and can be influenced based on the chitosan concentration, electrospinning conditions, and alignment mode. To test the usefulness of the fabricated scaffolds for 3D cell culture, a breast cancer cell line (MCF-7) was cultured on their surfaces and evaluated morphologically and biochemically. The obtained data showed a higher proliferation rate for cells grown on scaffolds compared to cells grown on two-dimensional adherent plates (tissue culture plate). The MTT assay revealed that the rate of cell proliferation on nanofibrous scaffolds is statistically significantly higher compared to tissue culture plate (P ≤ 0.001) after 14 days of culture. The formation of spheroids within the first few days of culture shows that the scaffolds effectively support 3D tissue culture from the outset of the experiment. Furthermore, 3D breast cancer tissues were spontaneously formed within 10 days of culture on aligned and non-aligned nanofibrous scaffolds, which suggests that the scaffolds imitate the in vivo extracellular matrix in the tumor microenvironment. Detailed mechanisms for the spontaneous formation of the 3D microtissues have been proposed. Our results suggest that scaffold surface topography significantly influences tissue formation and behavior of the cells.