We propose the narrowband perfect absorbers with enormously high fabrication tolerance, which consists of a low-contrast grating and a finite distributed Bragg reflector (DBR) layer with an ultrathin absorbing medium (graphene). It is numerically shown that the proposed perfect absorber outperforms the previously proposed schemes in fabrication tolerance. According to the rigorous coupled wave analysis (RCWA) and coupled mode theory (CMT) fitting, over a considerably wide range of grating width and thickness, the proposed absorber provides a proper ratio of leakage rate to loss rate while preserving resonant condition, so that almost perfect absorption (>99.9%) can be obtained. This result is attributed to the strong electric field confinement in the DBR region rather than the grating layer owing to lower index of grating compared to DBR. In addition, without degrading the fabrication tolerance, the bandwidth of the proposed absorber can be controlled by the DBR thickness (the number of pairs) and a narrow absorbing bandwidth of sub-nanometer is achieved with 8.5 Si/SiO2 pair stacked DBR.