The disordered quasi-one-dimensional magnet BaCu2SiGeO7 is considered as one of the best physical realizations of the random Heisenberg chain model, which features an irregular distribution of the exchange parameters and whose ground state is predicted to be the scarcely investigated random-singlet state (RSS). Based on extensive Si-29 NMR and magnetization studies of BaCu2SiGeO7, combined with numerical quantum Monte Carlo simulations, we obtain remarkable quantitative agreement with theoretical predictions of the random Heisenberg chain model and strong indications for the formation of a random-singlet state at low temperatures in this compound. As a local probe, NMR is a well-adapted technique for studying the magnetism of disordered systems. In this case, it also reveals an additional local transverse staggered field, which affects the low-temperature properties of the RSS. The proposed model Hamiltonian satisfactorily accounts for the temperature dependence of the NMR line shapes.