We present an extensive study of 162 early-type binary systems located in the LMC galaxy that show apsidal motion and have never been studied before. For the ample systems, we performed light curve and apsidal motion modelling for the first time. These systems have a median orbital period of 2.2 days and typical periods of the apsidal motion were derived to be of the order of decades. We identified two record-breaking systems. The first, OGLE LMC-ECL-22613, shows the shortest known apsidal motion period among systems with main sequence components (6.6 years); it contains a third component with an orbital period of 23 years. The second, OGLE LMC-ECL-17226, is an eccentric system with the shortest known orbital period (0.9879 days) and with quite fast apsidal motion period (11 years). Among the studied systems, 36 new triple-star candidates were identified based on the additional period variations. This represents more than 20% of all studied systems, which is in agreement with the statistics of multiples in our Galaxy. However, the fraction should only be considered as a lower limit of these early-type stars in the LMC because of our method of detection, data coverage, and limited precision of individual times of eclipses.