Constant false alarm rate (CFAR) is the desired property for automatic target detection in an unknown and non-stationary background. Here, a modified cell averaging CFAR (CA-CFAR) detector based on the Grubbs criterion (CAG-CFAR) is proposed for target detection in non-homogeneous background. The CFAR property of the CAG-CFAR with respect to the distribution parameter in exponential-distributed background is verified via Monte Carlo simulations. The detection performances of the proposed method in scenarios of multiple targets and clutter edges are investigated with different significance levels of the Grubbs criterion and sizes of reference window. Results show that the CAG-CFAR detector exhibits a similar detection performance as the CA-CFAR in homogenous environment with an appropriate significance level. At clutter edges, the CAG-CFAR detector attains a similar and acceptable false alarm rate control compared to several relevant competitors. In the multiple-target scenario, the proposed method achieves a robust detection performance with a low computational burden, whereas the competitors suffer performance degradations in varying degree. Simulations and experimental results verify the effectiveness and superiority of the proposed method in multiple-target situation with an unknown number of the interfering targets.