Romanian champions FC Cluj, Spartak Moscow, Lille and former European champions Celtic also went into the hat for Thursday's draw in Monaco after winning their ties.
Celtic, European Cup winners in 1967, beat Helsingborg 2-0 to complete a 4-0 aggregate win and leave Sweden without a team in the group stage for the 12th season in a row.
A first-half header from Pantelis Kapetanos sent Cluj into the group stage for the third time with a 1-0 win over FC Basel, who enjoyed home wins over Manchester United and Bayern Munich to reach the last 16 last season.
Spartak Moscow's 1-1 draw at Fenerbahce was enough to take them through while Lille needed an extra-time goal from Brazilian striker Tulio de Melo to overcome FC Copenhagen.
Monchengladbach, who have never played in the group stage, appeared to need a miracle as they travelled to Kiev, a 3-1 home defeat last week meaning they had to score three goals away from home to have a chance.
That suddenly began to look a possibility with two goals in eight minutes in the last quarter of the game against the three-times semi-finalists.
Kiev's jitters led to Yevhen Khacheridi turning a Juan Arango free-kick into his own goal in the 70th minute before the Venezuelan playmaker netted the second with a header.
The crowd were on tenterhooks until Nigerian Ideye Aide Brown rescued Kiev when he rushed clear of the defence and scored with a delicate chip over Marc-Andre ter Stegen in the 88th minute to give the Ukrainians a 4-3 aggregate win.
Cluj's only goal came when Greek striker Kapetanos met Diogo Valente's cross from the left with a powerful header in the 20th minute, although they were let off when Basel missed a penalty just before half-time.
Alexander Frei, all-time leading scorer for the Swiss national side, skied his effort well over the crossbar amid general bewilderment, leaving Switzerland without a team in the group stage for the first time since 2007/08.
HOSTILE CONDITIONS
Spartak, facing hostile conditions in Istanbul, quickly silenced the crowd in the sixth minute when Brazilian forward Ari swept in Aiden McGeady's low cross from the right to add to their 2-1 first-leg lead.
Senegalese forward Moussa Sow headed Fenerbahce level in the 69th minute to set up a dramatic finale but the Russians held on for a 3-2 aggregate win.
Gary Hooper put Celtic ahead on the half-hour, tapping in Giorgios Samaras's pass from close range as they dominated their second leg and Kenyan midfielder Victor Wanyama headed in the second following a corner in the 88th minute.
Lille, trailing 1-0 from the first leg against Copenhagen, levelled the aggregate score just before half-time when Ivorian forward Saloman Kalou crossed from the byline for Lucas Digne to fire into the roof of the net.
The winning goal came in the first period of extra time when Nolan Roux spun and crossed for Tulio to head home.
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