The first measurements of diboson production cross sections in proton-proton interactions at a center-of-mass energy of 5.02 TeV are reported. They are based on data collected with the CMS detector at the LHC, corresponding to an integrated luminosity of 302 pb-1. Events with two, three, or four charged light leptons (electrons or muons) in the final state are analyzed. The WW, WZ, and ZZ total cross sections are measured as σWW=37.0-5.2+5.5(stat)-2.6+2.7(syst) pb, σWZ=6.4-2.1+2.5(stat)-0.3+0.5(syst) pb, and σZZ=5.3-2.1+2.5(stat)-0.4+0.5(syst) pb. All measurements are in good agreement with theoretical calculations at combined next-to-next-to-leading order quantum chromodynamics and next-to-leading order electroweak accuracy.