The *one-shot success probability* of a noisy classical channel for transmitting one classical bit is the optimal probability with which the bit can be successfully sent via a single use of the channel. Prevedel *et al.* [Phys. Rev. Lett. 106, 110505 (2011)] recently showed that for a specific channel, this quantity can be increased if the parties using the channel share an entangled quantum state. In this paper, we characterize the optimal entanglement-assisted protocols in terms of the radius of a set of operators associated with the channel. This characterization can be used to construct optimal entanglement-assisted protocols for a given classical channel and to prove the limits of such protocols. As an example, we show that the Prevedel *et al.* protocol is optimal for two-qubit entanglement. We also prove some tight upper bounds on the improvement that can be obtained from quantum and nonsignaling correlations.