We present the apsidal motion and light-curve analyses of 21 eccentric eclipsing binaries located in the Small Magellanic Cloud. Most of these systems have never been studied before, hence their orbital and physical properties as well as the apsidal motion parameters are given here for the first time. All the systems are of early spectral type, having orbital periods up to 4 days. The apsidal motion periods were derived to be from 7.2 to 200 yr (OGLE-SMC-ECL-2194 having the shortest apsidal period among known main-sequence systems). The orbital eccentricities are usually rather mild (median of about 0.06), the maximum eccentricity being 0.33. For the period analysis using O-C diagrams of eclipse timings, in total 951 minima were derived from survey photometry as well as our new data. Moreover, six systems show some additional variation in their O-C diagrams, which should indicate the presence of hidden additional components in them. According to our analysis these third-body variations have periods from 6.9 to 22 yr.