Using density-functional studies and appropriate models, we show that the anomalous Hall effect (AHE) at the magnetic oxide interfaces can be tuned by modifying the Rashba interaction with the application of an external electric field. Explicit density-functional calculations were performed for the recently grown SrIrO3/SrMnO3 interface in presence of an electric field to demonstrate the effect. Our model calculation indicates the importance of off-diagonal inter-orbital hopping in driving a non-zero Berry curvature, the central quantity that leads to the AHE. Furthermore, with the broken inversion symmetry by an applied electric field, the Rashba interaction comes into play, which enhances the AHE, providing a mechanism to manipulate the AHE. The observed results from the model calculations are corroborated by the first-principles studies, which yield an anomalous Hall conductivity in reasonable agreement with the available experiments in absence of an electric field. Our work has potential applications in spintronics with possible use of AHE as a novel electric field sensor.