Romanian champions FC Cluj won 2-1 away to FC Basel, last season's conquerors of Manchester United, while Scottish champions Celtic earned a 2-0 victory at Helsingborg.
Spartak Moscow, who beat Fenerbahce 2-1, and FC Copenhagen, 1-0 winners over Lille, were the only home teams to win in Tuesday's five games.
Monchengladbach, beaten European Cup finalists in 1977, have not played in the elite club competition since the 1977/78 season, but got off to a flying start at home to the three-times semi-finalists from Ukraine.
Juan Arango produced a long diagonal pass which found Alexander Ring and he cut inside his marker to put the Bundesliga side ahead in the 13th minute.
But it was all downhill from there as Gladbach, playing their first competitive match of the season, were repeatedly undone by shoddy play.
Taras Mikhalik was allowed too much space before levelling with a deflected 25-metre shot in the 28th minute, and Andriy Yarmolenko put the visitors ahead eight minutes later, finishing clinically after Gladbach lost possession in midfield.
Arango created a flurry of chances in the second half, but it was Kiev who struck again with more help from the hosts.
Luuk de Jong deflected Miguel Veloso's free-kick into his own goal in the 81st minute, to leave the Germans needing to score at least three goals in next week's return leg.
SOUGOU DOUBLE
Basel dominated at home to Cluj for the first hour, with Marco Streller hooking them ahead spectacularly just before half-time.
But the Swiss double winners, hoping to qualify for the Champions League for the third season in a row, were undone as Senegalese forward Modou Sougou scored twice on the break in the space of six minutes during the second half.
Celtic took a second-minute lead at Helsingborg when Kris Commons, left unmarked at the far post, volleyed home Georgios Samaras' cross, before the Greece striker added the second himself with a 75th minute header.
Brazilian Cesar Santin's 38th-minute goal gave Copenhagen a 1-0 win over Lille, although the real hero for the Danes was goalkeeper Johan Wiland, who saved numerous chances included a Dimitri Payet penalty.
The game between Spartak Moscow and Fenerbahce featured three second-half goals in the space of 10 minutes.
Emmanuel Emenike opened the scoring for Spartak, only for Dirk Kuyt to draw the visitors level five minutes later.
However, a fine Dmitry Kombarov volley from Aiden McGeady's corner in the 69th ensured victory for the home side.
