The anticancer potential of quercetin (Q), a plant-derived flavonoid, and underlining molecular mechanisms are widely documented in cellular models in vitro. However, biomedical applications of Q are limited due to its low bioavailability and hydrophilicity. In the present study, the electrospinning approach was used to obtain polylactide (PLA) and PLA and polyethylene oxide (PEO)-based micro- and nanofibers containing Q, namely PLA/Q and PLA/PEO/Q, respectively, in a form of non-woven fabrics. The structure and physico-chemical properties of Q-loaded fibers were characterized by scanning electron and atomic force microscopy (SEM and AFM), X-ray powder diffraction (XRD), differential scanning calorimetry (DSC), goniometry and FTIR and Raman spectroscopy. The anticancer action of PLA/Q and PLA/PEO/Q was revealed using two types of cancer and nine cell lines, namely osteosarcoma (MG-63, U-2 OS, SaOS-2 cells) and breast cancer (SK-BR-3, MCF-7, MDA-MB-231, MDA-MB-468, Hs 578T, and BT-20 cells). The anticancer activity of Q-loaded fibers was more pronounced than the action of free Q. PLA/Q and PLA/PEO/Q promoted cell cycle arrest, oxidative stress and apoptotic cell death that was not overcome by heat shock protein (HSP)-mediated adaptive response. PLA/Q and PLA/PEO/Q were biocompatible and safe, as judged by in vitro testing using normal fibroblasts. We postulate that PLA/Q and PLA/PEO/Q with Q releasing activity can be considered as a novel and more efficient micro- and nano-system to deliver Q and eliminate phenotypically different cancer cells.