Let (Mn), (Nn) be two Hilbert-space-valued martingales adapted to some filtration (ℱn), with corresponding difference sequences (dn), (en), respectively. We assume that (Nn) weakly dominates (Mn), that is, for any convex non-decreasing function ϕ : ℝ+→ℝ+ and any n=1,2,… we have, almost surely, E(ϕ(|dn|)|ℱn−1)≤E(ϕ(|en|)|ℱn−1). We apply the Burkholder method to show that for a convex non-decreasing function Φ : ℝ+→ℝ+ satisfying some extra conditions we have, for any n=1,2,…, ‖Mn‖Φ≤CΦ‖Nn‖Φ, where ‖⋅‖Φ denotes an Orlicz norm with respect to Φ and CΦ is a constant which depends only on Φ. This approach unifies and extends the classical Burkholder inequalities for subordinated martingales and the inequalities for tangent martingales. The method leads to moment inequalities for Rosenthal-type dominated martingales and variance-dominated Gaussian martingales. All the constants obtained in the moment inequalities are of optimal order.