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We introduce the DESI LOW-Z Secondary Target Survey, which combines the wide-area capabilities of the Dark Energy Spectroscopic Instrument (DESI) with an efficient, low-redshift target selection method. Our selection consists of a set of color and surface brightness cuts, combined with modern machine learning methods, to target low-redshift dwarf galaxies ($z$ < 0.03) between $19 < r < 21$ with high completeness. We employ a convolutional neural network (CNN) to select high-priority targets. The LOW-Z survey has already obtained over 22,000 redshifts of dwarf galaxies (M$_* < 10^9$ M$_\odot$), comparable to the number of dwarf galaxies discovered in SDSS-DR8 and GAMA. As a spare fiber survey, LOW-Z currently receives fiber allocation for just ~50% of its targets. However, we estimate that our selection is highly complete: for galaxies at $z < 0.03$ within our magnitude limits, we achieve better than 95% completeness with ~1% efficiency using catalog-level photometric cuts. We also demonstrate that our CNN selections $z<0.03$ galaxies from the photometric cuts subsample at least ten times more efficiently while maintaining high completeness. The full five-year DESI program will expand the LOW-Z sample, densely mapping the low-redshift Universe, providing an unprecedented sample of dwarf galaxies, and providing critical information about how to pursue effective and efficient low-redshift surveys.