The establishment of a scalable scheme for quantum computing with addressable and long-lived qubits would provide a route to harnessing the laws of quantum physics to solve classically intractable problems. The design of many proposed platforms for quantum computing is driven by competing needs: isolating the quantum system from the environment to prevent decoherence, and easily and accurately controlling the system with external fields. For example, neutral-atom optical-lattice architectures provide environmental isolation through the use of states that are robust against fluctuating external fields, yet external fields are essential for qubit addressing. Here, we demonstrate the selection of individual qubits with external fields, while the qubits are in field-insensitive superpositions. We use a spatially inhomogeneous external field to map selected qubits to a different field-insensitive superposition, minimally perturbing unselected qubits, despite the fact that the addressing field is not spatially localized. We show robust single-qubit rotations on neutral-atom qubits located at selected lattice sites. This precise coherent control should be more generally applicable to state transfer and qubit isolation in other architectures using field-insensitive qubits.