We prove that an infinite block-Toeplitz matrix with finite diagonal support is totally nonnegative if and only if it is the weight matrix of a cylindrical network. This generalizes a well-known theorem of Brenti concerning finite totally nonnegative matrices and planar networks; in particular, our work gives an alternative, self-contained proof of the non-square case. Our argument employs Temperley-Lieb immanants, first introduced by Rhoades and Skandera, which are certain elements of Lusztig's dual canonical bases. As an application, we also obtain a new proof of a well-known theorem relating totally nonnegative block-Toeplitz matrices to interlacing polynomials.