The sides came into the second leg all square following a 1-1 draw in Kuwait last week, but 2013 runners-up Al Qadsia looked destined to exit the competition for much of the encounter.
Al Hadd are unbeaten at home in the competition this season and looked unlikely to give up that record when Akarandut Orok - the hosts' top AFC Cup goalscorer this year - gave them the lead in the 25th minute.
Ahmed Al Khattal doubled the advantage 11 minutes into the second half as Al Hadd looked set to become the first Bahraini side since Riffa in 2010 to reach the semi-finals.
Al Qadsia hit back through a Danijel Subotic penalty in the 68th minute after Bader Al Mutawa had been brought down in the box.
Despite seeing their opponents reduce the deficit, Al Hadd looked capable of seeing out the match and booking their place in the semi-finals.
However, a horror own goal from Mohammad Al Daoud with one minute of the 90 left to play ended their hopes of progression.
Al Qadsia will now face Indonesia's Persipura in the last four.
