We present the first measurement at the LHC of exclusive J/ψ photoproduction off protons, in ultraperipheral proton-lead collisions at √s(NN)=5.02  TeV. Events are selected with a dimuon pair produced either in the rapidity interval, in the laboratory frame, 2.5<y<4 (p−Pb) or −3.6<y<−2.6 (Pb−p), and no other particles observed in the ALICE acceptance. The measured cross sections σ(γ+p→J/ψ+p) are 33.2±2.2(stat)±3.2(syst)±0.7(theor)  nb in p−Pb and 284±36(stat)+27−32(syst)±26(theor)  nb in Pb−p collisions. We measure this process up to about 700 GeV in the γp center of mass, which is a factor of two larger than the highest energy studied at HERA. The data are consistent with a power law dependence of the J/ψ photoproduction cross section in γp energies from about 20 to 700 GeV, or equivalently, from Bjorken x scaling variable between ∼2×10−2 and ∼2×10−5, thus indicating no significant change in the gluon density behavior of the proton between HERA and LHC energies.