Network alignment aims to find the optimal nodes correspondences across two networks leveraging topological and/or attributes information. Existing methods of network alignment belong to two main categories: optimization-based methods and embedding-based methods. Optimization-based methods involve solving eigen-decomposition of a similarity matrix, solving quadratic assignment problem via sub-gradient optimization, or using heuristic-based iterative greedy match. Whereas the embedding-based methods utilize a cost function to learn node representation vectors in a latent space followed by efficient node matching. Optimization based methods are more accurate but they are computationally expensive. Embedding-based methods generally exhibit the opposite trend. In this paper, we propose SST-Align, which solves the network alignment task by using a hybrid approach. In the first step, SST-Align uses graphlet-count based topology signature of the vertices in an iterative greedy matching method for obtaining an initial alignment. Then in the second step, the initial alignment is used as self-supervised labels for learning node embedding by using a siamese neural network on top of a pair of graph convolutional networks. Extensive experiments conducted on six real-life graph alignment datasets demonstrate that our proposed method outperforms the current state-of-the-art graph alignment methods in terms of node mapping accuracy.