A non-intrusive reduced-basis (RB) method is proposed for parametrized unsteady flows. A set of reduced basis functions are extracted from a collection of high-fidelity solutions via a proper orthogonal decomposition (POD), and the coefficients of the reduced basis functions are recovered by a feedforward neural network (NN). As a regression model of the RB method for unsteady flows, the neural network approximates the map between the time/parameter value and the projection coefficients of the high-fidelity solution onto the reduced space. The generation of the reduced basis and the training of the NN are accomplished in the offline stage, thus the RB solution of a new time/parameter value can be recovered via direct outputs of the NN in the online stage. Due to its non-intrusive nature, the proposed RB method, referred as the POD-NN, fully decouples the online stage and the high-fidelity scheme, and is thus able to provide fast and reliable solutions of complex unsteady flows. To test this assertion, the POD-NN method is applied to the reduced order modeling (ROM) of the quasi-one dimensional Continuously Variable Resonance Combustor (CVRC) flow. Numerical results demonstrate the efficiency and robustness of the POD-NN method.