21 pages, 7 figuresThe effects of parton energy loss and pT-broadening in cold nuclear matter on the pT and centrality dependence, at various rapidities, of J/psi suppression in p-A collisions are investigated. Calculations are systematically compared to E866 and PHENIX measurements. The very good agreement between the data and the theoretical expectations further supports pT-broadening and the associated medium-induced parton energy loss as dominant effects in J/psi suppression in high-energy p-A collisions. Predictions for J/psi (and Upsilon) suppression in p-Pb collisions at the LHC are given