Magnetic systems have been extensively studied from both a fundamental physics perspective and as technological building blocks. The topological properties of magnonic excitations in these systems remain relatively unexplored, due to their inherently dissipative nature. The recent extension of the theory of topological classification to non-Hermitian Hamiltonians provides a pathway to engineer topological phases in dissipative systems. Here, we propose a magnonic realization of a topological, non-Hermitian system. A crucial ingredient of our proposal is the injection of spin current into the magnetic system, which alters and can even change the sign of terms describing dissipation. We show that the magnetic dynamics of an array of spin-torque oscillators can be mapped onto a non-Hermitian Su-Schrieffer-Heeger model exhibiting topologically protected edge states. The nontrivial topological phase is accessed by tuning the spin current injected into the array. We derive this result using both exact diagonalization of the effective non-Hermitian Hamiltonian and numerical analysis of the nonlinear equations of motion. In the nontrivial topological phase, a single spin-torque oscillator on the edge of the array is driven into auto-oscillation and emits a microwave signal, while the bulk oscillators remain inactive. Our findings have practical utility for memory devices and spintronics neural networks relying on spin-torque oscillators as constituent units.