Most targeting strategies of anticancer drug delivery systems (DDSs) rely on the surface functionalization of nanocarriers with specific ligands, which trigger the internalization in cancer cells via receptor-mediated endocytosis. The endocytosis implies the entrapment of DDSs in acidic vesicles (endosomes and lysosomes) and their eventual ejection by exocytosis. This process, intrinsic to eukaryotic cells, is one of the main drawbacks of DDSs because it reduces the drug bioavailability in the intracellular environment. The escape of DDSs from the acidic vesicles is, therefore, crucial to enhance the therapeutic performance at low drug dose. To this end, we developed a multifunctionalized DDS that combines high specificity towards cancer cells with endosomal escape capabilities. Doxorubicin-loaded mesoporous silica nanoparticles were functionalized with polyethylenimine, a polymer commonly used to induce endosomal rupture, and hyaluronic acid, which binds to CD44 receptors, overexpressed in cancer cells. We show irrefutable proof that the developed DDS can escape the endosomal pathway upon polymeric functionalization. Interestingly, the combination of the two polymers resulted in higher endosomal escape efficiency than the polyethylenimine coating alone. Hyaluronic acid additionally provides the system with cancer targeting capability and enzymatically controlled drug release. Thanks to this multifunctionality, the engineered DDS had cytotoxicity comparable to the pure drug whilst displaying high specificity towards cancer cells. The polymeric engineering here developed enhances the performance of DDS at low drug dose, holding great potential for anticancer therapeutic applications.