Measurement-induced phase transitions (MIPT) have attracted increasing attention due to the rich phenomenology of entanglement structures and their relation with quantum information processing. Since physical systems are unavoidably coupled to the environment, quantum noise, which can qualitatively modify or even destroy certain entanglement structures, needs to be considered in analyzing a system with MIPT. We investigate the effect of quantum noise modeled by a reset quantum channel acting on each site with a probability q on MIPT. Based on the numerical results from Clifford circuits, we show that the quantum noise can qualitatively change the entanglement properties - the entanglement obeys “area law” instead of “volume law” with a measurement rate p < pc. In the quantum noise-induced “area law” phase, the entanglement exhibits a novel q−1/3 power-law scaling. Using an analytic mapping from the quantum model to a classical statistical model, we further show that the “area law” entanglement is the consequence of the noise-driven symmetry-breaking field and the q−1/3 scaling can be understood as the result of Kardar-Parisi-Zhang (KPZ) fluctuations of the directed polymer with an effective length scale Leff ∼ q−1 in a random environment.