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The minimal gauged U(1)$_{L_μ-L_τ}$ model is a simple extension of the Standard Model and has a strong predictive power for the neutrino sector. In particular, the mass spectrum and couplings of heavy right-handed neutrinos are determined as functions of three neutrino Dirac Yukawa couplings, with which we can evaluate the baryon asymmetry of the Universe generated through their decay, i.e., leptogenesis. In this letter, we study leptogenesis in the minimal gauged U(1)$_{L_μ-L_τ}$ model. It turns out that the sign of the resultant baryon asymmetry for the case with the Dirac CP phase, $δ$, larger than $π$ is predicted to be opposite to that for $δ< π$. In addition, if lepton asymmetry is dominantly produced by the decay of the lightest right-handed neutrino, then the correct sign of baryon asymmetry is obtained for $δ> π$, which is favored by the current neutrino-oscillation experiments, whilst the wrong sign is obtained for $δ< π$. We further investigate a non-thermal leptogenesis scenario where the U(1)$_{L_μ-L_τ}$ breaking field plays the role of inflaton and decays into right-handed neutrinos, as a concrete example. It is found that this simple framework offers a successful inflation that is consistent with the CMB observation. We then show that the observed amount of baryon asymmetry can be reproduced in this scenario, with its sign predicted to be positive in most of the parameter space.