We present a version of the domination principle for the excessive measures of a general right process. The key point is that if we fix an excessive measure m, then any excessive measure absolutely continuous with respect to m admits a Radon-Nikodým derivative that is finely continuous off a suitable exceptional set. The domination principle is the statement that if the density of the potential of a measure \(\mu\) is dominated by the density of a second excessive measure \(\eta\), almost everywhere with respect to \(\mu\), then the potential of \(\mu\) is everywhere dominated by \(\eta\). This result sharpens earlier work of \textit{G. Mokobodzki} and the first named author, and it is related to the co- fine domination principle of K. Janssen. The principal tools used are the theory of Kuznetsov measures, and the ``essential limits'' pioneered by Chung and Walsh. Several applications of the domination principle are made. In particular we characterize the class of m-quasi-bounded potentials, extending work of Arsove and Leutwiler in Newtonian potential theory.