The development of the Automatic License Plate Recognition (ALPR) system has received much attention for the English license plate. However, despite being the sixth largest population around the world, no significant progress can be tracked in the Bengali language countries or states for the ALPR system addressing their more alarming traffic management with inadequate road-safety measures. This paper reports a computationally efficient and reasonably accurate Automatic License Plate Recognition (ALPR) system for Bengali characters with a new end-to-end DNN model that we call Bengali License Plate Network(BLPnet). The cascaded architecture for detecting vehicle regions prior to vehicle license plate (VLP) in the model is proposed to eliminate false positives resulting in higher detection accuracy of VLP. Besides, a lower set of trainable parameters is considered for reducing the computational cost making the system faster and more compatible for a real-time application. With a Computational Neural Network (CNN)based new Bengali OCR engine and word-mapping process, the model is characters rotation invariant, and can readily extract, detect and output the complete license plate number of a vehicle. The model feeding with17 frames per second (fps) on real-time video footage can detect a vehicle with the Mean Squared Error (MSE) of 0.0152, and the mean license plate character recognition accuracy of 95%. While compared to the other models, an improvement of 5% and 20% were recorded for the BLPnetover the prominent YOLO-based ALPR model and the Tesseract model for the number-plate detection accuracy and time requirement, respectively.