Brain tumor is one of the most aggressive diseases nowadays, resulting in a very short life span if it is diagnosed at an advanced stage. The treatment planning phase is thus essential for enhancing the quality of life for patients. The use of Magnetic Resonance Imaging (MRI) in the diagnosis of brain tumors is extremely widespread, but the manual interpretation of large amounts of images requires considerable effort and is prone to human errors. Hence, an automated method is necessary to identify the most common brain tumors. Convolutional Neural Network (CNN) architectures are successful in image classification due to their high layer count, which enables them to conceive the features effectively on their own. The tuning of CNN hyperparameters is critical in every dataset since it has a significant impact on the efficiency of the training model. Given the high dimensionality and complexity of the data, manual hyperparameter tuning would take an inordinate amount of time, with the possibility of failing to identify the optimal hyperparameters. In this paper, we proposed a Bayesian Optimization-based efficient hyperparameter optimization technique for CNN. This method was evaluated by classifying 3064 T-1-weighted CE-MRI images into three types of brain tumors (Glioma, Meningioma, and Pituitary). Based on Transfer Learning, the performance of five well-recognized deep pre-trained models is compared with that of the optimized CNN. After using Bayesian Optimization, our CNN was able to attain 98.70% validation accuracy at best without data augmentation or cropping lesion techniques, while VGG16, VGG19, ResNet50, InceptionV3, and DenseNet201 achieved 97.08%, 96.43%, 89.29%, 92.86%, and 94.81% validation accuracy, respectively. Moreover, the proposed model outperforms state-of-the-art methods on the CE-MRI dataset, demonstrating the feasibility of automating hyperparameter optimization.