Deep learning has emerged as the most promising approach in various fields; however, when the distributions of training and test data are different (domain shift), the performance of deep learning models can degrade. Semi-supervised domain adaptation (SSDA) is a major approach for addressing this issue, assuming that a fully labeled training set (source domain) is available, but the test set (target domain) provides labels only for a small subset. In this study, we propose a novel two-step momentum encoder-utilized SSDA framework, MoSSDA, for multivariate time-series classification. Time series data are highly sensitive to noise, and sequential dependencies cause domain shifts resulting in critical performance degradation. To obtain a robust, domain-invariant and class-discriminative representation, MoSSDA employs a domain-invariant encoder to learn features from both source and target domains. Subsequently, the learned features are fed to a mixup-enhanced positive contrastive module consisting of an online momentum encoder. The final classifier is trained with learned features that exhibit consistency and discriminability with limited labeled target domain data, without data augmentation. We applied a two-stage process by separating the gradient flow between the encoders and the classifier to obtain rich and complex representations. Through extensive experiments on six diverse datasets, MoSSDA achieved state-of-the-art performance for three different backbones and various unlabeled ratios in the target domain data. The Ablation study confirms that each module, including two-stage learning, is effective in improving the performance. Our code is available at https://github.com/seonyoungKimm/MoSSDA