We describe algorithms that detect 21cm line HI self-absorption (HISA) in large data sets and extract it for analysis. Our search method identifies HISA as spatially and spectrally confined dark HI features that appear as negative residuals after removing larger-scale emission components with a modified CLEAN algorithm. Adjacent HISA volume-pixels (voxels) are grouped into features in (l,b,v) space, and the HI brightness of voxels outside the 3-D feature boundaries is smoothly interpolated to estimate the absorption amplitude and the unabsorbed HI emission brightness. The reliability and completeness of our HISA detection scheme have been tested extensively with model data. We detect most features over a wide range of sizes, linewidths, amplitudes, and background levels, with poor detection only where the absorption brightness temperature amplitude is weak, the absorption scale approaches that of the correlated noise, or the background level is too faint for HISA to be distinguished reliably from emission gaps. False detection rates are very low in all parts of the parameter space except at sizes and amplitudes approaching those of noise fluctuations. Absorption measurement biases introduced by the method are generally small and appear to arise from cases of incomplete HISA detection. This paper is the third in a series examining HISA at high angular resolution. A companion paper (Paper II) uses our HISA search and extraction method to investigate the cold atomic gas distribution in the Canadian Galactic Plane Survey.
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