The Belarussian outfit came into Tuesday's clash a goal down from the first leg of their third-qualifying-round tie, and fell further behind early in the second.
Ibrahim Sidibe found the net from the penalty spot, but Debrecen were dealt a blow seven minutes before the break when Selim Bouadla was given his marching orders for a second bookable offence.
Aleksandr Volodko soon made Debrecen pay when he levelled the score on the night with an effort from outside the penalty area.
Vitali Rodionov then latched on to a curling cross from the left in the 66th minute to leave BATE trailing only on away goals, before Sergey Krivets rose well at the death to beat the opposition goalkeeper and nod his side into the play-off round.
Standard Liege overcame an early own goal to pick up a 2-1 win at Panathinaikos.
The tie was goalless heading into Tuesday's second leg, but Dino Arslanagic's 17th-minute error gave the home side the advantage.
Yanis Mbombo levelled nine minutes before the break and Paul-Jose M'Poku scored the decisive goal five minutes later.
The Belgians will be joined in the next round by Maribor, who staved off a Maccabi Tel Aviv fightback to register a 3-2 aggregate victory in Cyprus - a neutral venue being used due to unrest in Israel.
Agim Ibraimi reacted well to a parried effort to put Maribor, who held a one-goal advantage from last week, ahead in the 36th minute, before Tal Ben Haim and Eden Ben Basat made it 2-1 to Maccabi on the night.
However, Ibraimi soon dashed any hopes by finding the net for a second time in the 54th minute - one minute after Ben Basat's effort - to send Maribor through.
In the day's remaining clash, Lille successfully defended a two-goal advantage from the first leg by drawing 1-1 with Grasshopper.
Florent Balmont opened the scoring for the French club before Amir Abrashi levelled the second-leg score.
A red card for Daniel Pavlovic 22 minutes from time effectively ended the Swiss side's chances of mounting a comeback.
