Accurate influenza forecasting is essential for public health emergency preparedness and timely resource allocation. Although meteorological factors are established modulators of influenza transmission, existing deep-learning models rarely exploit this physical knowledge in a principled way. We introduce MeteoGST, a meteorology-driven spatiotemporal graph learning framework that combines (i) multi-scale feature extraction across seven operational meteorological variables (T_max, T_min, DTR, absolute humidity q, relative humidity RH, surface-pressure anomaly p_anom, and 10 m wind speed U10) via parallel dilated convolutions, TCN, and Transformer branches; (ii) a meteorology-aware dynamic graph attention network (MeteoGAT) whose edges blend geographic adjacency with time-varying meteorological similarity through a learned gate α; (iii) residual-trend decomposition with a peak-aware composite loss; and (iv) anti-smoothing meta-learning adaptation. On a 34-city pre-COVID-19 benchmark (2018–2019), MeteoGST achieves an RMSE of 0.65/0.85/1.12, MAE of 0.48/0.63/0.82, R2 of 0.84/0.76/0.70, and Peak F1 of 0.72/0.65/0.59 at 7-, 14-, and 30-day horizons, respectively—improvements of 4–6% over the strongest GNN baseline (MPNN-LSTM) and 31–44% over classical baselines (ARIMA/LSTM). Under a 2021–2022 distribution-shift stress test, the model retains its ranking at 1- and 4-week horizons (PCC 0.78/0.62) and degrades gracefully at 8 weeks, demonstrating robustness beyond the training distribution. MeteoGST offers an operationally deployable tool (MeteoGST-Lite: ≈1 ms per city-week on a laptop-class CPU) for integrated meteorology-aware influenza surveillance.