Time-dependent density-functional theory in the response regime is shown to yield qualitatively correct charge-transfer excitation energies in the system HeH(+) if the exact Kohn-Sham exchange potential is employed to determine the Kohn-Sham orbitals and eigenvalues entering the time-dependent density-functional calculation. The employed exact-exchange kernel is frequency-independent and, like conventional kernels in the local density approximation or in generalized gradient approximations, does not contribute to the charge-transfer excitation energy. This shows that it can be that not the exchange-correlation kernel, as generally believed, but the exchange-correlation potential plays the crucial role in the description of charge-transfer excitations.