We prove global-in-time Strichartz estimates for the shifted waveequations on non-trapping asymptotically hyperbolic manifolds. The key toolsare the spectral measure estimates from [Ann. Inst. Fourier, Grenoble 68(2018), pp. 1011–1075] and arguments borrowed from [Analysis PDE 9 (2016),pp. 151–192], [Adv. Math. 271 (2015), pp. 91–111]. As an application, weprove the small data global existence for any power p ∈ (1,1 + 4/n−1) for the shifted wave equation in this setting, involving nonlinearities of the form ±|u|p o r±|u|p−1u, which answers partially an open question raised in [Discrete Con-tin. Dyn. Syst. 39 (2019), pp. 7081–7099]. [ABSTRACT FROM AUTHOR]