The orbital period evolution of X-ray binaries provides fundamental clues to understanding mechanisms of angular momentum transfer and loss in these systems. We present an X-ray eclipse timing analysis of the transient low-mass X-ray binary AX J1745.6-2901. This system shows full eclipses and thus is one of the few objects of this class for which accurate orbital evolution studies can be carried out. We report on XMM-Newton and ASCA observations covering 30 complete X-ray eclipses spanning an interval of more than 20 yr. We improve the determination of the orbital period to a relative precision of 2 × 10-8, two orders of magnitudes better than previous measurements. We determine, for the first time, a highly significant rate of decrease of the orbital period dot{P}_{orb}=-4.03± 0.32{×} 10^{-11} s s-1. This is at least one order of magnitude larger than expected from conservative mass transfer and angular momentum losses due to gravitational waves and magnetic braking, and might result from either non-conservative mass transfer or magnetic activity changing the quadrupole moment of the companion star. Imprinted on the long-term evolution of the orbit, we observe highly significant eclipse leads-delays of ˜10-30 s, characterized by a clear state dependence in which, on average, eclipses occur earlier during the hard state.