We explore O(N) models in dimensions 4<d<6. Specifically, we investigate models of an O(N) vector field coupled to an additional scalar field via a cubic interaction. Recent results in d=6-ϵ have uncovered an interacting ultraviolet fixed point of the renormalization group (RG) if the number N of components of the vector field is large enough, suggesting that these models are asymptotically safe. We set up a functional RG analysis of these systems to derive three key results. First, we find that in d=5 the interacting fixed point exists all the way down to N=1. Second, we show that the standard O(N) universality classes are actually embedded in those of the cubic models, in that the latter exhibit the same values for (most of) the critical exponents, but feature an additional third RG relevant direction. Third, we address the critical question of global stability of the fixed-point potential to test whether the fixed point can underlie a viable quantum field theory.