Following up on the recent disovery of asymptotic safety and exact interacting UV fixed points in four-dimensional gauge theories coupled to matter, we investigate the impact of higher dimensional operators using the method of functional renormalisation. In the Veneziano limit, we establish that classically irrelevant couplings take well-defined interacting fixed point values of their own, despite of their non-renormalisability within standard perturbation theory. We also establish vacuum stability in the presence of higher dimensional scalar operators. Universal scaling exponents are found as well, showing that the higher order couplings remain parametrically irrelevant with near-Gaussian values. Our results provide a crucial consistency check for exact asymptotic safety of weakly coupled gauge theories. Similarities with fixed points in other theories including $4d$ quantum gravity are indicated.