Based on first-principles calculations, the electronic properties of a black AsP monolayer were investigated by applying an external strain. It was found that the electronic property and effective mass of black AsP monolayer exhibits strong anisotropy. Comparing with the armchair direction, the zigzag direction was more resistant to elastic deformation. When the strain was applied along the zigzag direction, a direct-indirect-direct transition of the band gap occurred with increasing tensile strain and compressive strain. The direct to indirect transition occurred at a −2% compressive strain when the strain was applied in the armchair direction. The direct-indirect transition occurred at biaxial compressive and tensile stains of −3% and 3%, respectively. In addition, the effective mass of electrons and holes in the armchair direction is an order of magnitude smaller than that associated with the zigzag direction, suggesting that electron and hole transport will occur preferentially along the armchair direction. Moreover, we found that the effective mass of the black AsP monolayer can be significantly tuned by strain.