We demonstrate synthetic azimuthal gauge potentials for Bose-Einstein condensates from engineering atom-light couplings. The gauge potential is created by adiabatically loading the condensate into the lowest energy Raman-dressed state, achieving a coreless vortex state. The azimuthal gauge potentials act as effective rotations and are tunable by the Raman coupling and detuning. We characterize the spin textures of the dressed states, in agreements with the theory. The lowest energy dressed state is stable with a 4.5-s half-atom-number-fraction lifetime. In addition, we exploit the azimuthal gauge potential to demonstrate the Hess-Fairbank effect, the analogue of Meissner effect in superconductors. The atoms in the absolute ground state has a zero quasi-angular momentum and transits into a polar-core vortex when the synthetic magnetic flux is tuned to exceed a critical value. Our demonstration serves as a paradigm to create topological excitations by tailoring atomlight interactions where both types of SO(3) vortices in the | F | = 1 manifold, coreless vortices and polar-core vortices, are created in our experiment. The gauge field in the stationary Hamiltonian opens a path to investigating rotation properties of atomic superfluids under thermal equilibrium.