A study is described of periodically perturbed primary bifurcations in two non-linearly coupled oscillators in both the non-resonant and resonant cases. The Krylov-Bogoliubov-Mitrolposky (KBM) averaging method is used to derive the bifurcation equations. The concept of normal form, singularity, unfolding theory, and theory of Z₂-symmetry bifurcations are employed to analyze the bifurcation equations and to enumerate the bifurcation diagrams in different components of bifurcation hypersurface in parameter space. The formation of several interesting patterns is presented. In the non-resonant case periodic perturbation is shown to have small effect on the primary bifurcations. The bifurcations in the subharmonic resonant case are Z₂-symmetry ones and they have codimension 3. On the other hand, the bifurcations in the main resonant case are shown to possess no symmetry properties and their codimension increases to 5. These bifurcation problems are analyzed in detail and the quasiglobal bifurcation classifications as well as bifurcation diagrams in the system are presented. It is shown that one can control vibration in non-linear systems with an appropriate choice of system parameters as suggested by some regions in the hypersurface where the amplitude of bifurcating solution is always zero. The study of periodically perturbed secondary bifurcation problems will be reported in a companion paper.