Period-one (P1) oscillations in a semiconductor laser under optical feedback from a narrowband fiber Bragg grating (FBG) are numerically investigated. FBG feedback enhances the stability of P1 oscillations compared to the conventional mirror feedback in the form of P1 microwave linewidth and phase noise reduction and residual noise peaks suppression. In the proposed scheme, the FBG has a narrow bandwidth smaller than the laser relaxation oscillation frequency. Then it effectively suppresses the coherence collapse of the laser by filtered feedback. Hence it can keep the laser in P1 operation even under relatively strong feedback. Besides, a uniform FBG has a comb-filtered reflectivity spectrum with a main lobe surrounded by several side lobes. Hence it can limit the external cavity modes by each lobe. As a result, FBG feedback can reduce microwave linewidth and phase noise by sustaining stronger feedback power and improve side-peak suppression ratio (SPSR) by filtering external cavity modes. The effects of stabilization are enhanced by properly increasing grating bandwidth. By fine-tuning the feedback delay time, the microwave linewidth can be reduced to a local minimum which reveals the optimal locking between P1 frequency and one of the external cavity modes. Increasing the feedback delay time, the local minimum linewidth can be further reduced. FBG feedback reduces the microwave linewidth by up to more than an order of magnitude and improves the SPSR by up to more than two orders of magnitude than mirror feedback using the same delay time.