We demonstrate the ability to control and shape the spectrotemporal content of extreme-ultraviolet (XUV) pulses produced by a seeded free-electron laser (FEL). The control over the spectrotemporal properties of XUV light was achieved by precisely manipulating the linear frequency chirp of the seed laser. Our results agree with existing theory, which allows us to retrieve the temporal properties (amplitude and phase) of the FEL pulse from measurements of the spectra as a function of the FEL operating parameters. Furthermore, we show the first direct evidence of the full temporal coherence of FEL light and generate Fourier limited pulses by fine-tuning the FEL temporal phase. The possibility of tailoring the spectrotemporal content of intense short-wavelength pulses represents the first step towards efficient nonlinear optics in the XUV to x-ray spectral region and will enable precise manipulation of core-electron excitations using the methods of coherent quantum control.