A new chiral amplification mechanism based on a stochastic approach is proposed. The mechanism includes five different chemical species, an achiral substrate (A), two chiral forms (L, D), and two intermediary species (LA, DA). The process occurs within a small, semipermeable compartment that can be diffusively coupled with the outside environment. The study considers two alternative primary sources for chiral species within the compartment, one chemical and the other diffusive. As a remarkable fact, the chiral amplification process occurs due to stochastic fluctuations of an intermediary catalytic species (LA, DA) produced in situ, given the interaction of the chiral species with the achiral substrate. The net process includes two different steps: the synthesis of the catalyst (LA and DA) and the catalytic production of new chiral species from the substrate. Stochastic simulations show that proper parameterization can induce a robust chiral state within the compartment regardless of whether the system is open or closed. We also show how an increase in the non-catalytic production of chiral species tends to negatively impact the homochirality degree of the system. By its conception, the proposed mechanism suggests a deeper connection with how most biochemical processes occur in living beings, a fact that could open new avenues for studying this fascinating phenomenon.