We study the scaling behavior of the Rényi entanglement entropy with smooth boundaries at the putative deconfined critical point separating the Néel antiferromagnetic and valence-bond-solid states of the two-dimensional J-Q_{3} model. We observe a subleading logarithmic term with a coefficient indicating the presence of four Goldstone modes, signifying the presence of an SO(5) symmetry at the transition point, which spontaneously breaks into an O(4) symmetry in the thermodynamic limit. This result supports the conjecture that an SO(5) symmetry emerges at the transition point, but reveals the transition to be weakly first-order. We demonstrate, through this Letter, a novel approach to detect emergent continuous symmetry and, more importantly, identify weakly first-order phase transitions efficiently, which have been notoriously challenging for conventional methods.