Quantum simulation on emerging quantum hardware is a topic of intense interest. While many studies focus on computing ground state properties or simulating unitary dynamics of closed systems, open quantum systems are a desirable target of study owing to their ubiquity and rich physical behavior. However, the non-unitary dynamics that makes these systems of interest is also difficult to simulate on near-term quantum hardware. Here, we report the digital quantum simulation of the dynamics of open quantum systems governed by a Lindblad equation using an adaptation of the quantum imaginary time evolution (QITE) algorithm. Our approach also allows the study of non-Markovian effects by simulation of time-convolutionless master equations and generalized master equations. We demonstrate our method through simulations of the spontaneous emission in the damped Jaynes-Cummings model on IBM quantum hardware. Our work demonstrates that the dynamics of open systems are accessible on near-term quantum hardware and without recourse to variational schemes. *HK and AJM are supported by NSF Grant Number 183920.