Historical maps are valuable resources that capture detailed geographical information from the past. However, these maps are typically available in printed formats, which are not conducive to modern computer-based analyses. Digitizing these maps into a machine-readable format enables efficient computational analysis. In this paper, we propose an automated approach to digitization using deep-learning-based semantic segmentation, which assigns a semantic label to each pixel in scanned historical maps. A key challenge in this process is the lack of ground-truth annotations required for training deep neural networks, as manual labeling is time-consuming and labor-intensive. To address this issue, we introduce a weakly-supervised age-tracing strategy for model fine-tuning. This approach exploits the similarity in appearance and land-use patterns between historical maps from neighboring time periods to guide the training process. Specifically, model predictions for one map are utilized as pseudo-labels for training on maps from adjacent time periods. Experiments conducted on our newly curated \textit{Hameln} dataset demonstrate that the proposed age-tracing strategy significantly enhances segmentation performance compared to baseline models. In the best-case scenario, the mean Intersection over Union (mIoU) achieved 77.3\%, reflecting an improvement of approximately 20\% over baseline methods. Additionally, the fine-tuned model achieved an average overall accuracy of 97\%, highlighting the effectiveness of our approach for digitizing historical maps.