Separability of the Dirac equation in Kerr-NUT-(A)dS spacetimes in all dimensions is intrinsically characterized. Namely, it is explicitly demonstrated that, in such spacetimes, there exists a complete set of first-order mutually commuting operators, one of which is the Dirac operator, that allows for common eigenfunctions which can be found in a separated form and correspond precisely to the general solution of the Dirac equation found by Oota and Yasui.