A measurement of the cross section for the combined two-photon production of charged pion and kaon pairs is performed using 1.2 fb-1 of data collected by the CLEO II detector at the Cornell Electron Storage Ring. The cross section is measured at invariant masses of the two-photon system between 1.5 and 5.0 GeV/c2, and at scattering angles more than 53° away from the γγ collision axis in the γγ center-of-mass frame. The large background of leptonic events is suppressed by utilizing the CsI calorimeter in conjunction with the muon chamber system. The reported cross section is compared with leading order QCD models as well as previous experiments. Although the functional dependence of the measured cross section disagrees with leading order QCD at small values of the two-photon invaraint mass, the data show qualitatively a transition to perturbative behavior at an invariant mass of approximately 2.5 GeV/c2.