We demonstrate the realization of supersymmetric quantum mechanics in the first-order Dirac oscillator equation by associating with it another Dirac equation, which may be considered as its supersymmetric partner. We show that both the particle and the antiparticle spectra, resulting from these two equations after filling the negative-energy states and redefining the physical ground state, indeed present the degeneracy pattern characteristic of unbroken supersymmetry. In addition, we analyze in detail two algebraic structures, each partially explaining the degeneracies present in the Dirac oscillator supersymmetric spectrum in the non-relativistic limit. One of them is the spectrum-generating superalgebra osp(2/2, ℝ), first proposed by Balantekin. We prove that it is closely connected with the supersymmetric structure of the first-order Dirac oscillator equation as its odd generators are the two sets of supercharges respectively associated with the equation and its supersymmetric partner. The other algebraic structure is an so(4)⊕so(3, 1) algebra, which is an extension of a similar algebra first considered by Moshinsky and Quesne. We prove that it is the symmetry algebra of the Dirac oscillator supersymmetric Hamiltonian. Some possible relations between the spectrum-generating superalgebra, the symmetry algebra, and their respective subalgebras are also suggested.