Density fluctuations near the QCD critical point can be probed via an intermittency analysis in relativistic heavy-ion collisions. We report the first measurement of intermittency in Au+Au collisions at s NN = 7.7-200 GeV measured by the STAR experiment at the Relativistic Heavy Ion Collider (RHIC). The scaled factorial moments of identified charged hadrons are analyzed at mid-rapidity and within the transverse momentum phase space. We observe a power-law behavior of scaled factorial moments in Au+Au collisions and a decrease in the extracted scaling exponent (ν) from peripheral to central collisions. The ν is consistent with a constant for different collisions energies in the mid-central (10-40%) collisions. Moreover, the ν in the 0-5% most central Au+Au collisions exhibits a non-monotonic energy dependence that reaches a minimum around s NN = 27 GeV. The physics implications on the QCD phase structure are discussed.