We present the discovery of the first millimeter afterglow of a short-duration γ-ray burst (SGRB) and the first confirmed afterglow of an SGRB localized by the GUANO system on Swift. Our Atacama Large Millimeter/Sub-millimeter Array (ALMA) detection of SGRB 211106A establishes an origin in a faint host galaxy detected in Hubble Space Telescope imaging at 0.7 ≲ z ≲ 1.4. From the lack of a detectable optical afterglow, coupled with the bright millimeter counterpart, we infer a high extinction, AV ≳ 2.6 mag along the line of sight, making this one of the most highly dust-extincted SGRBs known to date. The millimeter-band light curve captures the passage of the synchrotron peak from the afterglow forward shock and reveals a jet break at ${t}_{\mathrm{jet}}={29.2}_{-4.0}^{+4.5}$ days. For a presumed redshift of z = 1, we infer an opening angle, θjet = (15fdg5 ± 1fdg4), and beaming-corrected kinetic energy of $\mathrm{log}({E}_{{\rm{K}}}/\mathrm{erg})=51.8\pm 0.3$, making this one of the widest and most energetic SGRB jets known to date. Combining all published millimeter-band upper limits in conjunction with the energetics for a large sample of SGRBs, we find that energetic outflows in high-density environments are more likely to have detectable millimeter counterparts. Concerted afterglow searches with ALMA should yield detection fractions of 24%–40% on timescales of ≳2 days at rates of ≈0.8–1.6 per year, outpacing the historical discovery rate of SGRB centimeter-band afterglows.