Czerny-Turner (C-T) spectrometers are widely used in near-infrared spectral detection due to their compact structure, high stability and uniform dispersion. However, off-axis spherical use of mirrors introduce severe astigmatism, degrading broadband imaging quality, reducing energy utilization and limiting spectral resolution. To address this issue, a broadband composite astigmatism correction method based on diffraction control and a cylindrical lens is proposed. An off-axis parabolic mirror collimates the light source, and Fraunhofer single-slit diffraction from an asymmetric entrance slit produces controlled meridional divergence while maintaining sagittal collimation, regulating beam propagation in two orthogonal directions. Furthermore, a cylindrical lens is inserted between the focusing mirror and the detector to compensate for the residual sagittal astigmatism via its unidirectional focusing characteristic, achieving full-band astigmatism correction. Optical models for meridional and sagittal planes are established, analytical expressions for cylindrical lens parameters are derived, and an optical design is optimized for a 1 250~1 650 nm near-infrared spectrometer. Simulations show that compared with the traditional C-T structure, the proposed method reduces the sagittal and meridional spot root-mean-square(RMS) radius at 1 550 nm by 99.42% and 90.42%, respectively, significantly suppressing astigmatism. Prototype tests demonstrate spectral resolution better than 0.5 nm in the central band (near 1 550 nm) and 0.63 nm at the edge bands (1 310 nm, 1 610 nm), wavelength repeatability better than 0.3 nm, and a full-band scanning speed of 30.53 nm/s. This method is low-cost and highly engineerable, significantly improving broadband imaging quality while retaining the advantages of traditional C-T spectrometers, and provides an effective reference for designing high-performance near-infrared spectrometers.