This paper reports the measurement of J/ψ meson production in proton–proton (pp) and proton–lead (pPb) collisions at a center-of-mass energy per nucleon pair of 5.02 TeV by the CMS experiment at the LHC. The data samples used in the analysis correspond to integrated luminosities of 28 pb⁻¹ and 35 nb⁻¹ for pp and pPb collisions, respectively. Prompt and nonprompt J/ψ mesons, the latter produced in the decay of B hadrons, are measured in their dimuon decay channels. Differential cross sections are measured in the transverse momentum range of 2 < pT < 30 GeV/c, and center-of-mass rapidity ranges of |yCM| < 2.4 (pp) and −2.87 < yCM < 1.93 (pPb). The nuclear modification factor, R_pPb, is measured as a function of both pT and yCM. Small modifications to the J/ψ cross sections are observed in pPb relative to pp collisions. The ratio of J/ψ production cross sections in p-going and Pb-going directions, R_FB, studied as functions of pT and yCM, shows a significant decrease for increasing transverse energy deposited at large pseudorapidities. These results, which cover a wide kinematic range, provide new insight on the role of cold nuclear matter effects on prompt and nonprompt J/ψ production.