Density cumulant functional theory (DCFT) is one of a number of nascent electron correlation methods that are derived from reduced density matrices and cumulants thereof, instead of the wavefunction. Deriving properties from the density cumulant naturally yields methods that are size extensive and size consistent. In this work, we derive expressions for the analytic gradient, with respect to an external perturbation, for the DCFT-06 variant of density cumulant functional theory. Despite the fact that the DCFT-06 energy functional is stationary with respect to the density cumulant, the analytic gradients of the energy require the solution of perturbation-independent equations for both orbital and cumulant response. These two sets of linear response equations are coupled in nature and are solved iteratively with the solution of orbital and cumulant response equations each macroiteration, exhibiting rapid convergence. The gradients are implemented and benchmarked against coupled cluster theory with single and double excitations (CCSD) and CCSD with perturbative triple excitations [CCSD(T)], as well as accurate empirically corrected experimental data, for a test set comprising 15 small molecules. For most of the test cases, results from DCFT-06 are closer to CCSD(T) and empirical data than those from CCSD. Although the total energy and analytic gradient have the same asymptotic scaling, the present experience shows that the computational cost of the gradient is significantly lower.