In this paper, we propose a notion of high-order (zeroing) barrier functions that generalizes the concept of zeroing barrier functions and guarantees set forward invariance by checking their higher order derivatives. The proposed formulation guarantees asymptotic stability of the forward invariant set, which is highly favorable for robustness with respect to model perturbations. No forward completeness assumption is needed in our setting in contrast to existing high order barrier function methods. For the case of controlled dynamical systems, we relax the requirement of uniform relative degree and propose a singularity-free control scheme that yields a locally Lipschitz control signal and guarantees safety. Furthermore, the proposed formulation accounts for ``performance-critical" control: it guarantees that a subset of the forward invariant set will admit any existing, bounded control law, while still ensuring forward invariance of the set. Finally, a non-trivial case study with rigid-body attitude dynamics and interconnected cell regions as the safe region is investigated.