Protecting superconducting qubits from low-frequency noise is essential for advancing superconducting quan- tum computation. We here introduce a protocol for engineering dynamical sweet spots which reduce the sus- ceptibility of a qubit to low-frequency noise. Based on the application of periodic drives, the location of the dynamical sweet spots can be obtained analytically in the framework of Floquet theory. In particular, for the example of fluxonium biased slightly away from half a flux quantum, we predict an enhancement of pure- dephasing by three orders of magnitude. Employing the Floquet eigenstates as the computational basis, we show that high-fidelity single-qubit gates can be implemented while maintaining dynamical sweet-spot opera- tion. We further confirm that qubit readout can be performed by adiabatically mapping the Floquet states back to the static qubit states, and subsequently applying standard measurement techniques. Our work provides an in- tuitive tool to encode quantum information in robust, time-dependent states, and may be extended to alternative architectures for quantum information processing.