Exploiting B-meson decays for Standard Model tests and beyond requires a precise understanding of the strong final-state interactions that can be provided modelindependently by means of dispersion theory. This formalism allows one to deduce the universal pion-pion final-state interactions from the accurately known ππ phase shifts and, in the scalar sector, a coupled-channel treatment with the kaon-antikaon system. In this work an analysis of the decaysB 0 d → J/ψπ + π − andB 0 s → J/ψπ + π − is presented. We find very good agreement with the data up to 1.05 GeV in the ππ invariant mass, with a number of parameters reduced significantly compared to a phenomenological analysis. In addition, the phases of the amplitudes are correct by construction, a crucial feature for many CP violation measurements in heavy-meson decays.