This work proposes a selective estimator of harmonic current components based on a deep neural network (DNN), which is able to provide the amplitudes and phase shifts of these components through a quarter cycle of the current fundamental waveform. A sufficiently optimal configuration was reached for application in the harmonic estimation proposal from an exhaustive search for DNN parameters. The DNN training was performed from a set of current samples in the time domain. The evaluation test indicated that the DNN presents an average of approx. 99\% of amplitude errors smaller than 0.0036 pu and, in relation to the phase shifts, the average errors are smaller than 0.0041 rad. Furthermore, a case study targeting selective harmonic compensation by means of an active power filter is presented considering reference currents generated from the DNN estimations. The results show that there was a 59.3\% reduction in total harmonic distortion (THD) by using the proposed strategy, reducing from 29.88\% to 12.16\% which is still a high value, while individual (selected) harmonic components were attenuated into values between 80 and 94\%, indicating the viability of DNN in this type of application.