A vertical Bragg reflection waveguide (BRW) laser is proposed and simulated for reaching single-mode output, which operates at an exceptional point (EP). We study the properties of the EP-BRW laser using both the transfer matrix method and the traveling wave model. The mode selection mechanism for the EP-BRW laser and the key design parameters for optimum performance are analyzed. The effect of the EP on laser performance is investigated to achieve high confinement transversely of the optical field and low threshold gain. At the EP, such laser system exhibits exotic behaviors, such as high transverse gain, high single-mode stability, and high slope efficiency.