The Ivorians carried a one-goal deficit into the second leg of the second-round clash, but were back on terms in the tie by the 26th minute through Ibrahima Fofana.
Neither side was able to breach the opposing defence for the remainder of the half, but Adama Kangoute gave ASEC the overall lead three minutes into the second.
That left Constantine needing a goal to progress to the play-off round, but ASEC pounced to put the Algerian outfit to the sword.
Fofana grabbed a second, Foba Stevens Koffi netted twice and Adriel D'avila Ba Loua got in on the act as the hosts emphatically sealed the win.
Djoliba, the 2012 runners-up, left it late to overcome Wadi Degla, who held a 2-0 lead from the first leg.
The Egyptian side looked set to qualify until Djoliba mustered two goals with time ticking away to send the tie to a penalty shout-out, which they edged 5-4 to progress to the next round.
Tunisia are the most successful nation in the history of this competition and Bizertin could be in line to continue that trend following a 2-1 against Warri Wolves.
Edem Rjaibi found the net either side of a Kolawole Anubi strike to send Bizertin through.
Elsewhere, Abdul scored the only goal as Petro Luanda picked up a 1-0 aggregate triumph against Ismaily, while ZESCO United's 1-0 victory over Medeama was not enough to avoid a 2-1 overall defeat.
‘After Manchester City’s recent form, maybe they’re the underdogs against Manchester United!’ Former Red Devils defender on this weekend’s derby
‘Arteta, Alonso, Emery, me… none of us were physical players – we needed the understanding of the game. That probably helped us move into management’: Premier League boss reveals reasons for natural career progression
