New CCD observations for thirteen eccentric eclipsing binaries from the Large Magellanic Cloud were carried out using the Danish 1.54-meter telescope located at the La Silla Observatory in Chile. These systems were observed for their times of minima and 56 new minima were obtained. These are needed for accurate determination of the apsidal motion. Besides that, in total 436 times of minima were derived from the photometric databases OGLE and MACHO. The O − C diagrams of minima timings for these B-type binaries were analysed and the parameters of the apsidal motion were computed. The light curves of these systems were fitted using the program PHOEBE, giving the light curve parameters. We derived for the first time the relatively short periods of the apsidal motion ranging from 21 to 107 years. The system OGLE-LMC-ECL-07902 was also analysed using the spectra and radial velocities, resulting in masses 6.8, and 4.4 M ⊙ for the eclipsing components. For one system (OGLE-LMC-ECL-20112), the third-body hypothesis was also used for describing the residuals after subtraction of the apsidal motion, resulting in period of about 22 years. For several systems an additional third light was also detected, which makes these systems suspicious of triplicity.