In recent decades there has been a rapid development of methods to experimentally control individual quantum systems. A broad range of quantum control methods has been developed for two-level systems; however, the complexity of multilevel quantum systems make the development of analogous control methods extremely challenging. Here we exploit the equivalence between multilevel systems with SU(2) symmetry and spin-1/2 systems to develop a technique for generating new robust, high-fidelity, multilevel control methods. As a demonstration of this technique, we develop adiabatic and composite multilevel quantum control methods and experimentally realize these methods using a $171\mathrm{Yb}+$ ion system. We measure the average infidelity of the process in both cases to be around ${10}^{-4}$ , demonstrating that this technique can be used to develop high-fidelity multilevel quantum control methods and can, for example, be applied to a wide range of quantum computing protocols, including implementations below the fault-tolerant threshold in trapped ions.