We study the evolution of Tsallis entropy along the heat flow and establish concavity results in arbitrary dimensions. Extending earlier one-dimensional results, we prove that Tsallis entropy is concave along the heat flow for $q\in(0,3]$ in dimension one and for $q\in[1,3]$ in higher dimensions. The upper endpoint $q=3$ is sharp in every dimension. The proof is based on a nonlinear transformation of the heat equation, a sharp dimension-free functional inequality with constant $C_u=3$, and a rigorous justification of the integration-by-parts identities used in the argument. The sharp inequality is proved by an explicit integration-by-parts sum-of-squares identity, rather than by a computer-assisted semidefinite-programming search. As consequences, we recover a generalized de Bruijn identity, prove monotonicity of the associated $q$-Fisher information along the heat flow, and establish concavity results for Tsallis entropy power, including the Shannon entropy-power case and Costa's EPI as an endpoint. We also obtain an asymptotic entropy-power concavity statement for general initial data and a sharp auxiliary functional inequality which may be of independent analytic interest.