We determine the Landau-Zener transition probability in a dissipative environment including both longitudinal as well as transversal quantum-mechanical noise originating from a single noise source. For this, we use the numerically exact quasiadiabatic path integral, as well as the approximative nonequilibrium Bloch equations. We find that transversal quantum noise in general influences the Landau-Zener probability much more strongly than longitudinal quantum noise does at a given temperature and system-bath coupling strength. In other words, transversal noise contributions become important even when the coupling strength of transversal noise is smaller than that of longitudinal noise. We furthermore reveal that transversal noise renormalizes the tunnel coupling independent of temperature. Finally, we show that the effect of mixed longitudinal and transversal noise originating from a single bath cannot be obtained from an incoherent sum of purely longitudinal and purely transversal noise.