Simulating open quantum systems on digital quantum computers typically relies on the use of auxiliary qubits, resulting in overheads of noisy multi-qubit gates, that severely limit execution on near-term hardware. In this work, we explore the simulation of non-Markovian dynamics as well as memory channels leveraging the method of trajectory mixing, valid for mixed unitary channels. This allows to drastically reduce circuit depth, trading entangling gates for a statistical mixture of independent, pure state trajectories. Using a realistic noise model calibrated to modern quantum processors, we show the benefits of this approach, yielding higher state fidelity and better preservation of quantum correlations. This shows the possibility of simulating long-time non-Markovian evolutions with low noise and limited resources.