The phenomenon where a quantum system can be exponentially accelerated to its stationary state has been referred to as the Quantum Mpemba Effect (QMpE). Due to its analogy with the classical Mpemba effect, hot water freezes faster than cold water, this phenomenon has garnered significant attention. Although QMpE has been characterized and experimentally verified in different scenarios, the sufficient and necessary conditions to achieve such a phenomenon are still under investigation. In this paper, we address a sufficient condition for QMpE through a general approach for open quantum system dynamics. With the help of the Mpemba parameter introduced in this work to quantify how strong the QMpE can be, we discuss how our conditions can predict and explain the emergence of weak and strong QMpE in a robust way. As an application, by harnessing the intrinsic non-classical nature of squeezed thermal environments, we show how enhanced QMpE can be effectively induced when our conditions are met. We demonstrate that when the system interacts with thermal reservoirs, a hot qubit freezes faster than a cold qubit in the presence of squeezing. Our results provide tools and new insights, opening a broad avenue for further investigation at the most fundamental levels of this peculiar phenomenon in the quantum realm.