Time-series forecasting is complicated by nonlinear dynamics, regime transitions, and noise, while existing Topological Data Analysis (TDA) methods typically compute persistent homology independently across sliding windows, yielding static descriptors that fail to capture structural evolution. This study proposes N-BEATS+FZZ, a topology-enhanced forecasting framework that integrates FastZigzag persistence to track dynamic topological changes efficiently. Windowed segments are transformed into point clouds via time delay embedding, and window updates are encoded through vertex insertions and deletions, enabling the construction of a global zigzag persistence barcode from which both static features (Betti statistics, persistence amplitudes, topological entropy) and dynamic indicators (birth-death events, structural variation rates) are derived and incorporated into the N-BEATS architecture as covariates. Across 24 evaluation scenarios constructed from six benchmark cryptocurrency series, N-BEATS+FZZ achieves substantial accuracy gains, including an average 43.8% reduction in MAE and a 42.0% reduction in RMSE compared with classical TDA-based baselines. This demonstrates that modeling topological evolution yields more informative and robust representations for time-series forecasting.