We prove uniqueness in law for stochastic Volterra equations (SVEs for short) with completely monotone kernels and non-degenerate noise under mild regularity assumptions. In particular, our results reveal the regularization-by-noise effect for SVEs with singular kernels allowing for Hölder continuous multiplicative noise. In order to prove our results, we lift the SVE to an infinite dimensional stochastic evolution equation (SEE for short) defined on a Gelfand triplet of Hilbert spaces. We prove uniqueness in law for the lifted SEE by means of the generalized coupling approach, and then translate it into the original SVE.