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 sigma(WW) = 37:0(-5.2)(+5.5) (stat)(-2.6)(+2.7) (syst) pb, sigma(WZ) = 6.4(-2.1)(+2.5) (stat)(-0.3)(+0.5)(syst) pb, and sigma(ZZ) = 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.