Pyroelectricity is a very promising phenomenon in three- and two-dimensional materials, but first-principles calculations have not so far been used to elucidate the underlying mechanisms. Here we report density-functional theory (DFT) calculations based on the Born-Szigeti theory of pyroelectricity, by combining fundamental thermodynamics and the modern theory of polarization. We find satisfactory agreement with experimental data in the case of bulk benchmark materials, showing that the so-called electron-phonon renormalization, whose contribution has been traditionally viewed as negligible, is important. We predict out-of-plane pyroelectricity in the recently synthesized Janus MoSSe monolayer and in-plane pyroelectricity in the group-IV monochalcogenide GeS monolayer. It is notable that the so-called secondary pyroelectricity is found to be dominant in GeS monolayer. The present work opens a theoretical route to study the pyroelectric effect using DFT and provides a valuable tool in the search for new candidates for pyroelectric applications.