Dwarf carbon (dC) stars, main-sequence stars showing carbon molecular bands, are enriched by mass transfer from a previous asymptotic-giant-branch (AGB) companion, which has since evolved to a white dwarf. While previous studies have found radial-velocity variations for large samples of dCs, there are still relatively few dC orbital periods in the literature and no dC eclipsing binaries have yet been found. Here, we analyze photometric light curves from DR5 of the Zwicky Transient Facility for a sample of 944 dC stars. From these light curves, we identify 34 periodically variable dC stars. Remarkably, of the periodic dCs, 82% have periods less than two days. We also provide spectroscopic follow-up for four of these periodic systems, measuring radial velocity variations in three of them. Short-period dCs are almost certainly post-common-envelope binary systems, because the periodicity is most likely related to the orbital period, with tidally locked rotation and photometric modulation on the dC either from spots or from ellipsoidal variations. We discuss evolutionary scenarios that these binaries may have taken to accrete sufficient C-rich material while avoiding truncation of the thermally pulsing AGB phase needed to provide such material in the first place. We compare these dCs to common-envelope models to show that dC stars probably cannot accrete enough C-rich material during the common-envelope phase, suggesting another mechanism like wind-Roche lobe overflow is necessary. The periodic dCs in this paper represent a prime sample for spectroscopic follow-up and for comparison to future models of wind-Roche lobe overflow mass transfer.