A brain tumour is one of the major reasons for death in humans, and it is the tenth most common type of tumour that affects people of all ages. However, if detected early, it is one of the most treatable types of tumours. Brain tumours are classified using biopsy, which is not usually performed before definitive brain surgery. An image classification technique for tumour diseases is important for accelerating the treatment process and avoiding surgery and errors from manual diagnosis by radiologists. The advancement of technology and machine learning (ML) can assist radiologists in tumour diagnostics using magnetic resonance imaging (MRI) images without invasive procedures. This work introduced a new hybrid CNN-based architecture to classify three brain tumour types through MRI images. The method suggested in this paper uses hybrid deep learning classification based on CNN with two methods. The first method combines a pre-trained Google-Net model of the CNN algorithm for feature extraction with SVM for pattern classification. The second method integrates a finely tuned Google-Net with a soft-max classifier. The proposed approach was evaluated using MRI brain images that contain a total of 1426 glioma images, 708 meningioma images, 930 pituitary tumour images, and 396 normal brain images. The reported results showed that an accuracy of 93.1% was achieved from the finely tuned Google-Net model. However, the synergy of Google-Net as a feature extractor with an SVM classifier improved recognition accuracy to 98.1%.