The inelastic hadronic cross section in proton-lead collisions at a centre-of-mass energy pernucleon pair of 5.02 TeV is measured with the CMS detector at the LHC. The data sample,corresponding to an integrated luminosity of $ mathcal{L} =12.6 pm 0.4, mathrm{nb}^{-1}$, has been collected with an unbiased trigger for inclusive particle production. The cross section is obtained from the measured number of proton-lead collisions with hadronic activity produced in the pseudo rapidity $ eta $ ranges between 3 and 5 and/or $ -5 $ and $ -3 $, corrected for photon-induced contributions, experimental acceptance,and other instrumental effects. The inelastic cross section is measured to be $sigma_{ ext{inel}} ( ext{pPb}) = 2061 pm 3, (mathrm{stat}) pm 34, (mathrm{syst}) pm 72, (mathrm{lumi}), mathrm{mb}$. Various Monte Carlo generators, commonly used in heavy ion and cosmic ray physics, are found to reproduce the data within uncertainties. The value of $ sigma_{ ext{inel}} ( ext{pPb}) $ is compatible with that expected from the proton-proton cross section at 5.02 TeV scaled up within a simple Glauber approach to account for multiple scatterings in the lead nucleus, indicating that further net nuclear corrections are small.