The condensation of stiff, highly charged DNA molecules into compact structures by condensing agents ranging from multivalent ions to small cationic proteins is of major biological and therapeutic importance, yet the underlying microscopic mechanism remains poorly understood. It has been proposed that DNA condensation is a purely electrostatic phenomenon driven by the existence of a strongly correlated liquid (SCL) of counterions at the DNA surface. The same theoretical argument predicts that multivalent counterions overcompensate the DNA charge when present at high concentration, in turn destabilizing the condensates. Here, we demonstrate the occurrence of DNA charge inversion by multivalent ions through measurements of the electrophoretic mobility of condensed DNA. By observing the multivalent-ion-induced condensation of a single DNA molecule using magnetic tweezers, we further show that charge inversion influences condensation by modulating the barrier for condensate nucleation in a manner consistent with the SCL mechanism.