Biodegradable composite nanofibers were electrospun from poly(ε-caprolactone) (PCL) and poly(ethylene oxide) (PEO) mixtures dissolved in acetic and formic acids. The variation of PCL:PEO concentration in the polymer blend, from 5:95 to 75:25, revealed the tunability of the hydrolytic stability and mechanical properties of the nanofibrous mats. The degradation rate of PCL/PEO nanofibers can be increased compared to pure PCL, and the mechanical properties can be improved compared to pure PEO. Although PCL and PEO have been previously reported as immiscible, the electrospinning into nanofibers having restricted dimensions (250–450 nm) led to a microscopically mixed PCL/PEO blend. However, the hydrolytic stability and tensile tests revealed the segregation of PCL into few-nanometers-thin fibrils in the PEO matrix of each nanofiber. A synergy phenomenon of increased stiffness appeared for the high concentration of PCL in PCL/PEO nanofibrous mats. The pure PCL and PEO mats had a Young’s modulus of about 12 MPa, but the mats made of high concentration PCL in PCL/PEO solution exhibited 2.5-fold higher values. The increase in the PEO content led to faster degradation of mats in water and up to a 20-fold decrease in the nanofibers’ ductility. The surface of the PCL/PEO nanofibers was functionalized by an amine plasma polymer thin film that is known to increase the hydrophilicity and attach proteins efficiently to the surface. The combination of different PCL/PEO blends and amine plasma polymer coating enabled us to tune the surface functionality, the hydrolytic stability, and the mechanical properties of biodegradable nanofibrous mats.