Core-sheath fibers have numerous applications ranging from composite materials for advanced manufacturing to materials for drug delivery and regenerative medicine. Here, a simple and tunable approach for the generation of core-sheath fibers from immiscible solutions of dextran and polyethylene oxide is described. This approach exploits the entanglement of polymer molecules within the dextran and polyethylene oxide phases for free surface spinning into dry fibers. The mechanism by which these core-sheath fibers are produced after contact with a solid substrate (such as a microneedle) involves complex flows of the phase-separating polymer solutions, giving rise to a liquid-liquid core-sheath flow that is drawn into a liquid bridge. This liquid bridge then elongates into a core-sheath fiber through extensional flow as the contacting substrate is withdrawn. The core-sheath structure of the fibers produced by this approach is confirmed by attenuated total reflection Fourier-transform infrared spectroscopy and confocal microscopy. Tuning of the core diameter is also demonstrated by varying the weight percentage of dextran added to the reservoir from which the fibers are formed.