The triangular lattice of S=1/2 spins with XXZ anisotropy is a ubiquitous model for various frustrated systems in different contexts. We determine the quantum phase diagram of the model in the plane of the anisotropy parameter and the magnetic field by means of a large-size cluster mean-field method with a scaling scheme. We find that quantum fluctuations break up the nontrivial continuous degeneracy into two first-order phase transitions. In between the two transition boundaries, the degeneracy-lifting results in the emergence of a new coplanar phase not predicted in the classical counterpart of the model. We suggest that the quantum phase transition to the nonclassical coplanar state can be observed in triangular-lattice antiferromagnets with large easy-plane anisotropy or in the corresponding optical-lattice systems.