The Ghanaian championships crashed out of the elite continental inter-club championship 4-1 on aggregate due to a slim 1-0 home win in the first leg.
By the result at Stade 8 Mai 1945, Setif - 2014 champions - have progressed to the group stage, a phase the Ogya Lads were hoping to reach for the first time ever.
Mourad Benayad shot the home side into the lead in the 28th minute, before adding his second of the night on the stroke of half-time.
Television replays, however, showed the second goal was directed into the net by his hand on the blind side of Burkinabe referee Juste Ephrem Zio.
In the 62nd minute, the Black Eagles made it 3-0 through Abdelkader Bedrane, who finished off a neat move with a cool effort.
At this score, Setif were confirmed for the group stage, but that did not prevent them from adding one more goal in stoppage time.
Abdelhakim Amokrane reacted fast to slot a rebound into the net after goalkeeper Joseph Addo pushed out a free-kick.
Aduana overcame Libyan side Al-Tahaddy 2-1 on aggregate to make the first round.