Identifying qualitative changes in time-series data provides insights into dynamics behind the data. These qualitative changes can be detected through topological approaches, which first embed the time-series data into a high-dimensional space using a time-delay parameter and subsequently extract topological features, which describe a shape of data, from the embedded points. However, the essential topological features extracted at a single time-delay value are not sufficient for evaluating the qualitative changes, even when the well-chosen value is used. We propose a delay-variant embedding that constructs the extended topological features by regarding the time-delay as a variable parameter rather than as a single fixed value. The proposed method observes variations in the topological features where the time-delay serves as an extra dimension in the topological feature space. We theoretically prove that the proposed features are robust under the noise perturbation of time-series. We further combine the proposed topological features with the kernel technique in machine learning to classify general time-series data. We demonstrate the effectiveness of the method in the classification of synthetic noisy biological data and real electrocardiogram data. Our method outperforms the method based on a single time-delay value and, surprisingly, achieves the highest classification accuracy on average among conventional time-series analysis techniques.