For an electric vehicle (EV) with a continuously variable transmission (CVT), a novel convex programming (CP)-based co-design method is proposed to minimize the total-cost-of-ownership (TCO). The integration of the electric machine (EM) and the CVT is the primary focus. The optimized system with co-design reduces the TCO by around 5.9% compared to a non-optimized CVT-based EV (based on off-the-shelf components) and by around 2% compared to the EV equipped with a single-speed transmission (SST). By taking advantage of the control and design freedom provided by the CVT, the optimal CVT, EM and battery sizes are found to reduce the system cost. It simultaneously finds the optimal CVT speed ratio and air-flow rate of the cooling system reducing the energy consumption. The strength of co-design is highlighted by comparing to a sequential design, and insights into the design of a low-power EV that is energy-efficient and cost-effective for urban driving are provided. A highly integrated EM-CVT system, which is efficient, low-cost and lightweight, can be expected for future EV applications.