The qubit Rabi oscillations are known to be non-decaying (though with a fluctuating phase) if the qubit is continuously monitored in the weak-coupling regime. In this paper we propose an experiment to demonstrate these persistent Rabi oscillations via low-frequency noise correlation. The idea is to measure a qubit by two detectors, biased stroboscopically at the Rabi frequency. The low-frequency noise depends on the relative phase between the two combs of biasing pulses, with a strong increase of telegraph noise in both detectors for the in-phase or anti-phase combs. This happens because of self-synchronization between the persistent Rabi oscillations and measurement pulses. Almost perfect correlation of the noise in the two detectors for the in-phase regime and almost perfect anticorrelation for the anti-phase regime indicates a presence of synchronized persistent Rabi oscillations. The experiment can be realized with semiconductor or superconductor qubits.