3D-printed hydrogels are particularly advantageous as drug-delivery platforms but their loading with water-soluble active compounds remains a challenge requiring the development of innovative inks. Here, we propose a new 3D extrusion-based approach that, by exploiting the internal gelation of the alginate, avoids the post-printing crosslinking process and allows the loading of epirubicin-HCl (EPI). The critical combinations of alginate, calcium carbonate and d-glucono-δ-lactone (GDL) combined with the scaffold production parameters (extrusion time, temperature, and curing time) were evaluated and discussed. The internal gelation in tandem with 3D extrusion allowed the preparation of alginate hydrogels with a complex shape and good handling properties. The dispersion of epirubicin-HCl in the hydrogel matrix confirmed the potential of this self-crosslinking alginate-based ink for the preparation of 3D-printed drug-delivery platforms. Drug release from 3D-printed hydrogels was monitored, and the cytotoxic activity was tested against MCF-7 cells. Finally, the change in the expression pattern of anti-apoptotic, pro-apoptotic, and autophagy protein markers was monitored by liquid-chromatography tandem-mass-spectrometry after exposure of MCF-7 to the EPI-loaded hydrogels.