We study the effect of shared non-signaling correlations for the problem of simulating a channel using noiseless communication in the one-shot setting. For classical channels, we show how to round any non-signaling-assisted simulation strategy\,---\,which exactly corresponds to the meta-converse for channel simulation\,---\,to a strategy that only uses shared randomness. For quantum channels, we round any non-signaling-assisted simulation strategy to a strategy that only uses shared entanglement. As our main result, we prove a guarantee on the ratio of success probabilities of at least $(1-\tfrac{1}{\mathbf{e}})$, for both the classical and the quantum setting. We further show this ratio to be optimal. It can be improved to $(1-\tfrac{1}{t})$ using $O(\ln \ln(t))$ additional bits (qubits) of communication.