The modern convolutional neural networks although achieve great results in solving complex computer vision tasks still cannot be effectively used in mobile and embedded devices due to the strict requirements for computational complexity, memory and power consumption. The CNNs have to be compressed and accelerated before deployment. In order to solve this problem the novel approach combining two known methods, low-rank tensor approximation in Tucker format and quantization of weights and feature maps (activations), is proposed. The greedy one-step and multi-step algorithms for the task of multilinear rank selection are proposed. The approach for quality restoration after applying Tucker decomposition and quantization is developed. The efficiency of our method is demonstrated for ResNet18 and ResNet34 on CIFAR-10, CIFAR-100 and Imagenet classification tasks. As a result of comparative analysis performed for other methods for compression and acceleration our approach showed its promising features.