Exoskeletons can assist the daily life activities of the elderly with weakened muscle strength, but traditional rigid exoskeletons bring parasitic torque to the human joints and easily disturbs the natural movement of the wearer’s upper limbs. Flexible exoskeletons have more natural human-machine interaction, lower weight and cost, and have great application potential. Applying assist force according to the patient’s needs can give full play to the wearer’s remaining muscle strength, which is more conducive to muscle strength training and motor function recovery. In this paper, a design scheme of an elbow exoskeleton driven by flexible antagonistic cable actuators is proposed. The cable actuator is driven by a nonlinear series elastic mechanism, in which the elastic elements simulate the passive elastic properties of human skeletal muscle. Based on an improved elbow musculoskeletal model, the assist torque of exoskeleton is predicted. An assist-as-needed (AAN) control algorithm is proposed for the exoskeleton and experiments are carried out. The experimental results on the experimental platform show that the root mean square error between the predicted assist torque and the actual assist torque is 0.00226 Nm. The wearing experimental results also show that the AAN control method designed in this paper can reduce the activation of biceps brachii effectively when the exoskeleton assist level increases.