Penetrative turbulence, which occurs in a convectively unstable fluid layer and penetrates into an adjacent, originally stably stratified layer, is numerically and theoretically analyzed. As example we pick the canonical Rayleigh-Bénard geometry, but now with the bottom plate temperature Tb>4∘C, the top plate temperature Tt≤4∘C, and the density maximum around Tm≈4∘C in between, resulting in penetrative turbulence. Next to the Rayleigh number Ra, the crucial new control parameter as compared to standard Rayleigh-Bénard convection is the density inversion parameter θm(Tm-Tt)/(Tb-Tt). The crucial response parameters are the relative mean midheight temperature θc and the overall heat transfer (i.e., the Nusselt number Nu). We numerically show (for Ra up to 1010) and theoretically derive that θc(θm) and Nu(θm)/Nu(0) are universally(i.e., independently of Ra) determined only by the density inversion parameter θm and succeed to derive these universal dependences. In particular, θc(θm)=(1+θm2)/2, which holds for θm below a Ra-dependent critical value, beyond which θc(θm) sharply decreases and drops down to θc=1/2 at θm=θm,c. This critical density inversion parameter θm,c can be precisely predicted by a linear stability analysis. Finally, we numerically identify and discuss rare transitions between different turbulent flow states for large θm.