The period-change rate (PCR) of pulsating variable stars is a useful probe of changes in their interior structure, and thus of their evolutionary stages. So far, the PCRs of classical Cepheids in the Large Magellanic Cloud (LMC) have been explored in a limited sample of the total population of these variables. Here, we use a template-based method to build observed-minus-computed (O - C) period diagrams, from which we can derive PCRs for these stars by taking advantage of the long time baseline afforded by the Digital Access to a Sky Century @Harvard light curves, combined with additional data from the Optical Gravitational Lensing Experiment, the MAssive Compact Halo Object project, Gaia's Data Release 2, and in some cases the All-Sky Automated Survey. From an initial sample of 2315 sources, our method provides an unprecedented sample of 1303 LMC classical Cepheids with accurate PCRs, the largest for any single galaxy, including the Milky Way. The derived PCRs are largely compatible with theoretically expected values, as computed by our team using the Modules for Experiments in Stellar Astrophysics code, as well as with similar previous computations available in the literature. Additionally, five long-period (P > 50 d) sources display a cyclic behaviour in their O - C diagrams, which is clearly incompatible with evolutionary changes. Finally, on the basis of their large positive PCR values, two first-crossing Cepheid candidates are identified.