It is proved that pointwise commuting formally normal operators which are dominated by a single essentially normal operator are essentially normal and essentially spectrally commuting. The question when essential normality of a polynomial in an operator implies essential normality of that operator is solved in this way. Furthermore, domination by essentially normal powers of formally normal operators are studied and, as a consequence, extended versions of Nelson's criterion for essential spectral commutativity are proposed. Subsequent domination results ensuring joint subnormality of systems of operators are proved. Several applications to multidimensional moment problems are found.