As a promising polymer for the production of biomaterials and drug delivery systems, poly(lactic acid) (PLA) is characterized by its relative hydrophobicity, as well as its chemical and biological inertness. Here, we aimed to improve the biological properties of PLA-based materials via the covalent attachment of a hydrophilic biocompatible glycopolymer, namely poly(2-deoxy-N-methacrylamido-D-glucose) (PMAG) on their surface. PMAG is a water-soluble polymer that contains glucose units in its side chains, which are responsible for good biocompatibility and the ability to attach bioactive molecules. In the developed protocol, PMAG was synthesized by controlled radical polymerization in the presence of a reversible addition–fragmentation chain transfer (RAFT) agent, followed by the conversion of glycopolymer terminal dithiobenzoate functionality into a primary amino group (PMAG-NH2). PLA-based films served as model aliphatic polyester materials for developing the surface biofunctionalization protocol. According to that, PMAG-NH2 covalent immobilization was carried out after alkali treatment, allowing the generation of the surface-located carboxyl groups and their activation. The developed modification method provided a one-point attachment of hydrophilic PMAG to the hydrophobic PLA surface. PMAG samples, which differed by the degree of polymerization, and the variation of polymer concentration in the reaction medium were applied to investigate the modification efficacy and grafting density. The developed single-point polymer grafting approach provided the efficient functionalization with a grafting density in the range of 5–23 nmol/cm2. The neat and modified polymer films were characterized by a number of methods, namely atomic force microscopy, thermogravimetric analysis, ellipsometry, and contact angle measurements. In addition, an ArgGlyAsp-containing peptide (RGD peptide) was conjugated to the PMAG macromolecules grafted on the surface of PLA films. It was shown that both surface modification with PMAG and with PMAG-RGD peptide enhanced the adhesion and growth of mesenchymal stem cells as compared to a neat PLA surface.