We exploit echo-enabled harmonic generation (EEHG) to produce fully coherent free-electron laser (FEL) pulses at soft-x-ray wavelengths and shape their spectrotemporal content. In an EEHG FEL, the longitudinal phase space of the relativistic electron beam that amplifies light is precisely tailored using two external seed lasers and two magnetic chicanes. We show that the spectrotemporal properties of the emitted radiation can be controlled by tuning the bandwidth, linear frequency chirp, and intensity of one of the seed lasers. The experimental data are supported by analytical and numerical models. Our results open a pathway toward coherent control of quantum processes at short wavelengths in the fields of applied physics, chemistry and biology, where manipulating the radiation spectrum is essential. The ability to precisely control the spectrotemporal content of intense, short-wavelength FEL pulses and the low sensitivity of the radiation to electron-beam imperfections make the technique an ideal candidate for use in chirped-pulse amplification schemes.