The ArgoNeuT Collaboration presents the first measurements of inclusive muon neutrino charged current differential cross sections on argon. Obtained in the NuMI neutrino beam line at Fermilab, the flux-integrated results are reported in terms of outgoing muon angle and momentum. The data are consistent with the Monte Carlo expectation across the full range of kinematics sampled, 0°<θ(μ)<36° and 0<P(μ)<25 GeV/c. Along with confirming the viability of liquid argon time projection chamber technology for neutrino detection, the measurements allow tests of low-energy neutrino scattering models important for interpreting results from long baseline neutrino oscillation experiments designed to investigate CP violation and the orientation of the neutrino mass hierarchy.