The general relativistic precession rate of periastra in close-in exoplanets can be orders of magnitude larger than the magnitude of the same effect for Mercury. The realization that some of the close-in exoplanets have significant eccentricities raises the possibility that this precession might be detectable. We explore in this work the observability of the periastron precession using radial velocity and transit light curve observations. Our analysis is independent of the source of precession, which can also have significant contributions due to additional planets and tidal deformations. We find that precession of the periastra of the magnitude expected from general relativity can be detectable in timescales of less than or similar to 10 yr with current observational capabilities by measuring the change in the primary transit duration or in the time difference between primary and secondary transits. Radial velocity curves alone would be able to detect this precession for supermassive, close-in exoplanets orbiting inactive stars if they have similar to 100 data points at each of two epochs separated by similar to 20 yr. We show that the contribution to the precession by tidal deformations may dominate the total precession in cases where the relativistic precession is detectable. Studies of transit durations with Kepler might need to take into account effects arising from the general relativistic and tidal induced precession of periastra for systems containing close-in, eccentric exoplanets. Such studies may be able to detect additional planets with masses comparable to that of Earth by detecting secular variations in the transit duration induced by the changing longitude of periastron.