Recent advances in the functional material and biomedical applications of nanorods call for a fundamental understanding of the active motion of nanorods in a viscoelastic medium. Molecular dynamics simulations are performed to investigate a model system consisting of force-driven active thin nanorods in a melt of unentangled polymers. The activeness of a thin nanorod arises from a constant external force applied uniformly along the rod. The simulations demonstrate that the active force overcomes the randomness of the diffusive motion and results in a ballistic motion along the direction of the applied force at long timescales. The constant speed of the force-driven ballistic motion is determined by the balance of the active force and the friction from the coupling of the nanorod with the polymer viscosity. The friction coefficient, which is computed as the ratio of the active force and the speed, decreases as the active force increases. The origin of the reduction in the friction coefficient is the high speed that allows the nanorod to renew its local environment faster than the relaxation time of melt chains. A scaling theory is developed to quantify the dependence of the friction coefficient on the strength of the active force. The simulations also demonstrate that the force-driven ballistic motion suppresses the rotational diffusion of the rod and cuts off the de-correlation of the rod axis with time. On the scaling level, the long-time trajectory of a force-driven active nanorod piercing through unentangled polymers may be described as a stretched array of "active blobs", where the short-time random-walk trajectory within an active blob is unperturbed by the active force.