We present a novel deep learning method to separately extract the two-dimensional flux information of the foreground galaxy (deflector) and background system (source) of Galaxy-Galaxy Strong Lensing events using U-Net (GGSL-Unet for short). In particular, the segmentation of the source image is found to enhance the performance of the lens modeling, especially for ground-based images. By combining mock lens foreground+background components with real sky survey noise to train the GGSL-Unet, we show it can correctly model the input image noise and extract the lens signal. However, the most important result of this work is that the GGSL-UNet can accurately reconstruct real ground-based lensing systems from the Kilo Degree Survey (KiDS) in one second. We also test the GGSL-UNet on space-based (HST) lenses from BELLS GALLERY, and obtain comparable accuracy of standard lens modeling tools. Finally, we calculate the magnitudes from the reconstructed deflector and source images and use this to derive photometric redshifts (photo-z), with the photo-z of the deflector well consistent with spectroscopic ones. This first work, demonstrates the great potential of the generative network for lens finding, image denoising, source segmentation, and decomposing and modeling of strong lensing systems. For the upcoming ground- and space-based surveys, the GGSL-UNet can provide high-quality images as well as geometry and redshift information for precise lens modeling, in combination with classical MCMC modeling for best accuracy in the galaxy-galaxy strong lensing analysis.