We study the quantum Zeno effect (QZE) induced by continuous partial measurement in the presence of short-correlated noise in the system Hamiltonian. We study the survival probability and the onset of the QZE as a function of the measurement strength and find that, depending on the noise parameters, the quantum Zeno effect can be enhanced or suppressed by the noise in different regions of parameter space. Notably, the conditions for the enhancement of the QZE are different when determined by the short-time or long-time behavior of the survival probability, or by the measurement strength marking the onset of the quantum Zeno regime.