I am not quite sure about the reference :( I always thought of this fact as follows.

Matrix elements of tensor powers of a representation U are all possible monomials in matrix elements of U, so the space of all their linear combinations are values of all possible polynomials in the matrix elements of U. Now, by definition of H, values of matrix elements of U separate elements of G/H, so every function on G/H (including all irreducible characters) can be written as a polynomial in the matrix elements of U in the case of finite groups, or can be approximated by polynomials with arbitrary precision in the case of compact infinite groups and the ground field being R or C (Stone-Weierstrass).

Now, to complete the proof, we may use orthogonality of matrix elements: if E_{ij} are matrix elements of an irrep V, and F_{ij} --- matrix elements of an irrep W (all thought of as functions on the group), then for the standard bilinear form on the ring of functions C(G) we have (E_{ij},F_{kl})=0 unless V is isomorphic to W and, in the latter case, i=l, j=k (in which case the value is 1) - here I probably want the order of the group to not be divisible by char(k) in the finite case, or the group to be compact, and the field be real/complex in the infinite case. Since irreducible characters can be approximated by polynomials in matrix elements, such a character cannot be orthogonal to all matrix elements of tensor powers and is, therefor, contained in one of them.