We present a new gauge/gravity duality construction of the Klebanov-Strassler throat which takes corrections to the anomalous mass dimension proposed in  into account on the gauge theory side and both the dilaton and the axion run on the gravity side. The corresponding supergravity solutions are found using equations for type IIB flows with N=1 supersymmetry obtained in . We find that magnetic couplings of the axion to D7-branes filling 4-d spacetime and wrapping 4-cycles at locations of duality transitions and invisible Dirac 8-branes whose worldvolume emanates from the worldvolume of the D7-branes are the sources for the runnings of the dilaton and the axion. Our construction provides the first explicit example of a gauge/gravity duality mapping with a running dilaton or a running axion which is an important component towards finding gravity duals to gauge theories with physically more interesting renormalization group flows such as pure confining gauge theories in four dimensions. The D7-branes also serve as gravitational source for Seiberg duality transitions. The supergravity background has distinct features which could be useful for constructing cosmological models and studying possibilities for probing stringy signatures from the early universe.