Advanced biomaterials and scaffolds for tissue engineering place high demands on materials and exceed the passive biocompatibility requirements previously considered acceptable for biomedical implants. Together with degradability, the activation of specific cell–material interactions and a three-dimensional environment that mimics the extracellular matrix are core challenges and prerequisites for the organization of living cells to functional tissue. Moreover, although bioactive signalling combined with minimization of non-specific protein adsorption is an advanced modification technique for flat surfaces, it is usually not accomplished for three-dimensional fibrous scaffolds used in tissue engineering. Here, we present a one-step preparation of fully synthetic, bioactive and degradable extracellular matrix-mimetic scaffolds by electrospinning, using poly(D,L-lactide-co-glycolide) as the matrix polymer. Addition of a functional, amphiphilic macromolecule based on star-shaped poly(ethylene oxide) transforms current biomedically used degradable polyesters into hydrophilic fibres, which causes the suppression of non-specific protein adsorption on the fibres’ surface. The subsequent covalent attachment of cell-adhesion-mediating peptides to the hydrophilic fibres promotes specific bioactivation and enables adhesion of cells through exclusive recognition of the immobilized binding motifs. This approach permits synthetic materials to directly control cell behaviour, for example, resembling the binding of cells to fibronectin immobilized on collagen fibres in the extracellular matrix of connective tissue.