We explore the association of non-neutralized currents with solar flare occurrence in a sizable sample of observations, aiming to show the potential of such currents in solar flare prediction. We use the regularly produced high quality vector magnetograms by the Helioseismic Magnetic Imager and more specifically the Space weather HMI Active Region Patches (SHARP). Through a newly established method, that incorporates detailed error analysis, we calculate the non-neutralized currents contained in active regions (AR). Two predictors are produced, namely the total and the maximum unsigned non-neutralized current. Both are tested in AR time-series and a representative sample of point-in-time observations during the interval 2012-2016. The average values of non-neutralized currents in flaring active regions are by more than an order of magnitude higher than in non-flaring regions and correlate very well with the corresponding flare index. The temporal evolution of these parameters appears to be connected to physical processes, such as flux emergence and/or magnetic polarity inversion line formation that are associated with increased solar flare activity. Using Bayesian inference of flaring probabilities, it is shown that the total unsigned non-neutralized current outperforms significantly the total unsigned magnetic flux and other well established current-related predictors. Thus, it shows good prospects for inclusion in an operational flare forecasting service. We plan to use the new predictor in the framework of the FLARECAST project along with other highly performing predictors.