The Deans, who played the final half-hour of normal time with 10-men after seeing Jack Lee sent-off, went down 4-1 in the shoot-out after the second round qualifying tie had finished 1-1 after 90 minutes.
Substitute Liam Kimber saw his spot-kick saved by goalkeeper James Mayo, who netted Brockenhurt's winning penalty, while Ben Anderson's effort from 12 yards struck the crossbar - striker Zak Willett was the only Horndean player to find the net from the spot.
Earlier, the Deans had taken the lead through Chad Field 10 minutes after the interval, with his clever flicked effort from Willett's cross finding the bottom corner.
But the hosts had Lee dismissed on the hour mark after he picked up a second yellow for bringing down Christie Ward in the area. Wessex Premier leading scorer Silvano Obeng sent home the resulting penalty.
Despite a man disadvantage, Horndean remained on top and could have won it in normal time.
Manager Michael Birmingham replaced goalkeeper Cameron Scott with stopper Leon Pitman with penalties looming, but the change could not inspire the Deans to a shoot-out victory.
Horndean: Scott; Taw, Field, Lee, Miller, Hookey, Howes, Anderson, Tierney, Willett, Duffin. Subs: Emeny (Tierney, 61), Kimber (Hookey, 88), Pitman (Scott, 90), Hooper, Brownlie.