The change in the argument of periastron of eclipsing binaries, i.e., the apsidal motion caused by classical and relativistic effects, can be measured from variations in the difference between the time of minimum light of the primary and secondary eclipses. Poor apsidal motion rate determinations and large uncertainties in the classical term have hampered previous attempts to determine the general relativistic term with sufficient precision to test General Relativity predictions. As a product of the TESS mission, thousands of high-precision light curves from eclipsing binaries are now available. Using a selection of suitable well-studied eccentric eclipsing binary systems, we aim to determine their apsidal motion rates and place constraints on key gravitational parameters. We compute the time of minimum light from the TESS light curves of 15 eclipsing binaries with precise absolute parameters and with an expected general relativistic contribution to the total apsidal motion rate greater than 60%. We use the changing primary and secondary eclipse timing differences over time to compute the apsidal motion rate, when possible, or the difference between the linear periods as computed from primary and secondary eclipses. For a greater time baseline we carefully combine the high-precision TESS timings with archival reliable timings. We determine the apsidal motion rate of 9 eclipsing binaries, 5 of which are reported for the first time. From these, we are able to measure the general relativistic apsidal motion rate of 6 systems with sufficient precision to test General Relativity for the first time using this method. This test explores a regime of gravitational forces and potentials that had not been probed earlier. We find perfect agreement with the theoretical predictions, and we are able to set stringent constraints on two parameters of the parametrised post-Newtonian formalism.
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