The study of apsidal motion in eccentric eclipsing binaries provides an important observational test of theoretical models of stellar structure and evolution. New ground-based and space-based photometric data have been obtained and archival spectroscopic measurements were used in this study of three detached early-type and southern-hemisphere eccentric eclipsing binaries GM Nor (P=1.88d, e=0.05), V397 Pup (3.00d, 0.30), and PT Vel (1.80d, 0.12). Their TESS observations in several sectors have also been included and the corresponding light curves were solved using the PHOEBE code. As a result, new accurate photoelectric times of minimum light have been obtained. The newly completed O-C diagrams were analyzed using all reliable timings found in the literature and calculated using the TESS light curves. New or improved values for the elements of apsidal motion were obtained. Using ESO archive spectroscopy, for V397 Pup, the precise absolute parameters were newly derived: M1=3.076(35)M_{sun}_, M2=2.306(35)M_{sun}_, and R1=2.711(55)R_{sun}_, R2=1.680(55)R_{sun}_. For PT Vel the absolute dimensions were improved: M1=2.204(25)M_{sun}_, M2=1.638(25)M_{sun}_, and R1=2.108(30)R_{sun}_, R2=1.605(30)R_{sun}_. For GM Nor, the less accurate absolute parameters based on the light curve analysis were evaluated: M1=1.94(15)M_{sun}_, M2=1.84(14)M_{sun}_, and R1=2.27(20)R_{sun}_, R2=2.25(20)R_{sun}_. We found more precise and relatively short periods of apsidal motion of about 80, 335, and 160 years, along with the corresponding internal structure constants, log k2, -2.524, -2.361, and -2.563, for GM Nor, V397 Pup, and PT Vel, respectively. Relativistic effects are small but not negligible, making up to 10% of the total apsidal motion rate in all systems. No marks of the presence of the third body were revealed in the light curves, on the O-C diagrams, or in the reduced spectra of the eccentric systems studied here.