Radio detection of cosmic rays and neutrinos is an established technique pursued by many experiments. However, current reconstruction methods ignore bin-to-bin correlations of the measured waveforms, which limits reconstruction resolution and, so far, has prevented calculations of event-by-event uncertainties. In this work, we solve this shortcoming and present a likelihood description of neutrino and cosmic-ray signals in radio detectors. Using simulated data, we demonstrate that with this method, parameters such as the neutrino energy and direction, or lower-level parameters such as the energy fluence, can be obtained more precisely and with correct event-by-event uncertainties. The reconstruction code will be available through the open-source software NuRadioReco.