We construct the functional integral of Abelian Chern-Simons theory with toral gauge group T=t/Λ≅ U(1)ⁿ at level K, where K:Λ×Λ→Z is an even, integral, nondegenerate symmetric bilinear form, by exact zeta-regularized Gaussian evaluation of the formal quotient integral over connections modulo gauge. For closed 3-manifolds, this yields a topological invariant; for manifolds with boundary, the relative functional integral produces the canonical boundary state. The resulting theory satisfies the required axioms of a (2+1)-dimensional TQFT.